There are lots of rumors, heresay, impressions and not a lot of hard tech data about why old gear may sound different or better or worse than a new piece of gear. There can be marketing fluff as well, perhaps telling people what they want to hear to drive sales of something.
In some cases there are documented differences in materials or manufacturing methods or specs. I'll try to call those out where I think I can support it although unless I can measure something concrete I won't offer a sonic decision or opinion. Too many of those out there already. There is no shortage of youtube gurus who want you to hear their magical "results" through all of the audio compression and computer speakers (or ear buds). To say nothing of how it was recorded.
Speakers
Here you have magnets, baskets and then soft parts. It is really the soft parts that are tricky to duplicate. The cones, spiders and surrounds etc. are often an experimental mix of things until you get what you are looking for. The cones in particular were not some standard material, the different manufacturers had their own witches brews for them and many of these firms are gone. How do I know this? I consulted on getting Weber Speakers off tghe ground. And Ted Weber's brother in-law was Kent Pyle, of the Pyle speakers family and back in the day you might have known them as Utah. They contract manufactured speakers for Fender, Vox and others. So there was some experience there.
Gone too are the old growth trees and their long, dense fibers which affects the cones. So while you can get a 12" basket and bolt an N-size AlNiCo V magnet to it with the correct size gap, getting the rest of that speaker to sound like a P12N or a Celestion G12 can be a real challenge. On the plus side there aren't a million options to choose from and the soft parts are not very expensive. But you really have to be ready to experiment to find the combos of parts and treatment that give you what you're looking for. Sweating the details is the game here.
Thankfully there are more speaker makers today than there have been in a very long time and they produce a wonderful range of models to try out. It is a very good thing that we (and our music!) are not hostage to irreplaceable 60+ year old speakers that are disintegrating. Because that is what 60 year old speakers do. As they say in boxing, "Father time has never lost a fight". It goes for lots of other things as well.
Transformers
There are 3 big pieces here - the core laminations, the insulation and the wire. I suppose you can add a fourth - whether the transformer is impregnated, and with what. I used wax on the first run of OEI because it can give a little bit of protection for overheating in some cases. Its used in some missile systems to protect electronics because it absorbs heat needed to melt it which keeps other things cooler just a little longer. While substances melt, the energy absorbed goes into changing phase and the temp stays flat. Probably like "the stall" in BBQ but I haven't done the reading there to prove that. (anyone?)
The problem was wax made a real mess, so I switched to electrical varnish. I didn't notice a difference in sound or response. I did notice that I did not spend hours with a heat gun cleaning the things up.
The core materials have changed a good bit, and modern materials while "better" can tend to not be what we want in a guitar amp. Modern materials tend to be more linear and have a more narrow hysteresis curves, so they are "better" in a strict linear sense but "better" is not necessarily what we want if we are imitating a part from the 1950's or 60's. In some cases I was able to use modern stuff, in some cases I had to find the "worst" spec modern material to get me closer to what would have been on hand way back when. Different alloys with different annealing can have very different saturation flux levels, very different transfer curves and so on. That has a big effect on how the overall unit responds in the circuit. You can mix and match materials in a core to get different characteristics, you can try different lacing approaches and even play with the thickness of the core to adjust for some factors. Net net, I initially thought that using something like M6 laminations in designs would be "better" and in nearly all the cases it was somewhere between "different" and "not as good". The devices were not designed with it in mind and that probably shouldn't be a big surprise.
The wire has changed with respect to actual diameter (which affects turns per layer etc) and also what it is insulated with. When making the OEI units, Simcha had to adjust the windings a bit to maintain the proper leakage inductance & capacitance while dealing with wire that fit differently. The DCR of the windings also can matter, particularly in power transformers where I^2 * R losses can (in theory) become sag on the B+ rail. When your 100W Marshall is pulling 400mA or whatever at full tilt, if your B+ is say 500V you could be dropping 40V if your winding resistance is 100 ohms. (just a made up example) Can you hear that or feel that in your playing? I don't know.
The insulation has changed - for the better. I have heard of old Vox outputs that literally had newsprint inside them. It is important to remember that England had a pretty rough time in WW-II and the 1950's and even early 1960s were a time of scrappy cost-effective solutions with what was on hand. There was a lot of rebuilding to do. Often the answer is like the 'holy grail' from Indiana Jones - not a cup crusted with diamonds, but that banged up rusty thing in the corner that got the job done.
One reason OEI used Nomex is because it is very close to paper but can handle extreme heat. We did the same thing at Weber Speakers, replacing paper voice coils with Nomex so as to not impact the motor geometry but to make them more reliable. The upside is they tend to be very reliable, the downside is the cost. There ain't no free lunch.
Pots
Here the basic answer is pretty simple - there have been lots of tapers over the years and they were offered because they had utility in certain situations. A classic example is the oscillator in a tremolo circuit; many schematics (such as Fender) called for a reverse audio taper on the pot. This was to make the change in speed consistent as you turned the control, and to have it turn in an intuitive direction to get faster or slower. There were also some tapped pots used for treble in the EQ section of the "brown" era Fender amplifiers. I suspect they stopped to simplify and lower costs. Over time using more generic parts saved money for the pot manufacturers and the musical instrument makers. The mfr gets to make bigger runs and lower costs, and the MI companies get to buy and inventory fewer parts, and you have a simpler assembly process if mixing up parts has no effect.
For guitars, having a 10% taper, where about half of your resistance is from about 7-10 on the knob, makes the control a lot more "playable" by the performer. You brush the volume knob down and play your rhythm part, and then a quick tweak and you're taking a lead. It is very convenient and surprisingly effective.
Amplifier volume and tone controls can have specific tapers that have a concrete impact on which sounds happen where on the dials - and if they happen at all. You can wind up with specific values too close together to be achievable in some cases. Try putting a linear volume pot into a guitar amp and see what happens. In theory, all of the values should be there - but in practice it's really not so clear.
Here is a great chart of some various tapers that I grabbed from a post by Bruce Egnater. I have some smaller & less complete diagrams below it. This one is a little blurry but gives you a better view of how complex the topic actually is. You can still call up CTS and order custom pots but you need to usually order a bunch to make it worth everyone's while. I priced pots out with Ted Weber back in the day and they became affordable around 1,000pc. Can you say, "Commitment"?
Other taper diagrams:
Amps seem to wind up with the 20% taper and guitars seem to wind up with a 10% to work their best. YMMV. Always test and see what you like. Just turn the control to "5" (halfway) and measure the resistance. If it is a 500K pot, you should see ~50K if it's a 10% taper, ~100K if it is a 20% taper and so on. It's not super precise - or difficult - to figure out what it probably is.
If you are doing a repair, try to make sure you know the taper of the pot you are replacing and match it as best you can.
Resistors
Old carbon comp resistors had a few interesting wrinkles. For one, they were often rated at higher operating voltages, for another they would add subtle amounts of distortion depending on the operating voltage and voltage swing. The immortal RG Keene put his lens to this and it probably should be required reading. (see below)
Wire
Many a great story starts with "so I know a guy who"... and in this case, he built some proof of concept amps for Fender to demonstrate why wire was their issue in the later "silverface" amps. He doesn't talk about it anymore because of internet trolling, which stinks. But the point is that the older amps all used single conductor cloth braided wire. This did 2 big things: 1 - it stayed where you put it so lead dress could be more consistent and 2 - it had a single eddy current around it, being a single conductor. While that 2nd bit may sound a bit tweaky - perhaps a lot - the reality is all of these amps are unstable at some frequency, usually some high frequency. Signals in the various leads need to be kept apart or crossed at 90 degrees to minimize interactions. All wires have fields around them realting to current flow. And you always have interactions. High frequencies have harmonics, and they can be sub-harmonics. So your output section instability at 40K could turn into some kind of ringing or modulation at 20k, 10K, or 5K, etc. Net net, his demo for Fender involved re-rewiring an amp with the old wire and voila, it sounded like the old ones. Due to various factors though Fender could not use that wire in production and had to move on. In the same vein, the old Marshalls used 7-stranded wire and that impacted how they behaved. I would love to see more careful study of things like this to ID what is (or perhaps is not) going on. I am not a believer in 'magic', but wire pops up a lot of places as a factor - including transformers and guitar pickups. The gauges have changed as has the insulation. Saying "it does/doesn't" matter without data is not productive. It seems intuitive that different materials would have different impacts - but would they be audible? Would they be material? (no pun intended) Seems worth a look. I did re-wire most of my old SF Pro Reverb with single conductor cloth braid wire and it is warmer sounding to my ears now but I have no measurements. It is on the "to do" list though. And even with measurements I haven't accounted for wire positioning. Or insulation material effects. A physicist might focus on skin effect and high frequencies traveling on the wire (where multi-strand or "Litz" wire is much better), but these are far higher than anything in our guitar amps (aren't they?)
Tubes
The fundamentals of a tube "type" are established in its physical components, configurations and locations. So what can be different? Why do people get religious sometimes about this 12AX7 or that one?
Things like the cathode coatings, plate construction and so on can vary and possibly have audible effects. Then there are things like the overall size of the plate - it can keep the same geometry (eg plate curves) but be physically larger. How do these kinds of details affect the sound? Technically they perhaps should not, but we are often running strange modes in the world of Guitar. Clipping a lot, usually. I have seen some well intentioned but useless "reviews" where someone shoots a video and ascribes all kinds of magic to some specific model. What I have not seen is any actual data to back that up. A good start would be a spectral analysis, there are easy (and cheap) ways to pull this off, and of course you need a decent sample size and all that grown up stuff.
A typical approach in recording engineering is to take a track (a test tone or whatever) and put that on one track. Then run it into the thing in question - plugin, outboard device, whatever and record that output on a 2nd track. Now flip the phase on one of them and add them together. The original signal should cancel and you'll be left with what the device/software "added". This is a poor man's THD meter in effect. If you use a single test tone you can do a spectral analysis and see the harmonic series produced. We could do this for resistors & caps of course as well.
Pet theory? Part of it is that you're hearing differences in mu from tube to tube. They are not identical, they are hand assembled to tight tolerances but then they get sorted into ranges of noise and gain to get sold ideally at a premium for some desirable combination of things. Newer examples may even have different transfer characteristics.Testing some of this out is also on the "to do" list, ideally with a reseller partner because purchasing enough tubes to perform a proper study could easily cost thousands.
Capacitors
Want to start a fight? Talk about capacitors with a bunch of guitar amp junkies. The basics are pretty simple: separate two plates with an insulator. The insulator will have characteristics which affect the signal, and the sizes of the plates themselves will relate to total capacitance. If you look in component catalog, it's easy to see that there are multiple parts available with the same voltage and capacitance values. Why? Intuitively there must be differences that make this worthwhile. Are they audible? Well, now you have a brawl on your hands.
A really good read about how they are made and how they work is featured in one of my favorite electronic texts, "Valve Amplifiers", by Morgan Jones. Chapter 4 focuses on component technology, including transformers, tubes, capacitors and more. (see link below) Mr Jones actually spent time designing capacitors for a living, so he has some very interesting light to shed on the whole subject. TL;DR? They are microphonic, they add distortion and phase changes and their values are not constant with frequency. Can you hear it? I don't know, but feel free to knock yourselves out.
Not literally, of course.
References / Tools
Morgan Jones - Valve Amplifiers https://www.amazon.com/Valve-Amplifiers-Morgan-Jones/dp/0080966403
RG Keene Investigates CC resistors: http://www.geofex.com/article_folders/carbon_comp/carboncomp.htm
12AX7 THD Measurements (comparison but with a too small sample size): https://www.vtadiy.com/home/tube-amplifiers/il-terzo-12ax7-el84/12ax7-vacuum-tube-comparison/
Eddy Currents https://en.wikipedia.org/wiki/Eddy_current#
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