"Do different power transformers actually change the tone of an amp?"
Yes, they can.
Keeping it simple, if PT-A puts out X voltage and Y current
under load for heater, secondaries, and any other accessory taps, and PT-B puts out
higher or
lower voltages and currents, then there could be a difference in both sound and feel of the amp.
Not only that, but a transformer produces a giant magnetic field. And how a transformer is constructed and the quality of materials both affect this magnetic field. That in turn affects things like how hot or cool the transformer runs or how much hum the transformer injects into other parts of the circuit.
Even if PT-A and PT-B put out the
exact same voltages and currents under the same operating conditions, PT-B might induce audible hum into the amp or even vibrate and hum mechanically, while PT-A might not, depending on how they're each constructed.
Some transformers use higher quality laminations like M6 or copper flux bands, which affect things a bit. Then you've got the quality of the potting/lacquering, quality of the wiring (leading to longevity and resistance to shorting under spicy operating conditions), and etc.
Two transformers can supply the same
idle voltage and current for their various output taps. But
under load, while you're playing, they can each respond and react differently due to any number of factors. e.g. you're chugging away at high volume with some super sick Swedish death metal. PT-A keeps up, supplying the appropriate power. But PT-B starts to struggle, failing to immediately provide the necessary current and/or voltages—leading to sag.
At the end of the day, choose quality transformers from reputable companies and you'll be ok. Avoid bargain-bin, no-name iron.
It just determines the plate voltage, right?
No. They do much more than that and they are not the only thing that contributes to an amp's plate voltage at idle or under load.
Reason I ask is I don't hear much of a difference in those old IIC+ amps. Whether it's a 105 or export...?? Seems to sound the same to me. But they're preamp distortion based amps so... IDK??
This is a slightly different scenario. You're asking about the same amp model that came from the manufacturer with two different transformers. In that case, I would trust the manufacturer to hopefully know what they're doing, so both amps sound the same.
That said, in vintage Marshalls like '70s – '80s JCM800s, they came with any number of different power transformers. And they didn't all supply the same voltages and current. Some yielded B+ as high as 530v and some yielded B+ as low as 380v. Mesa was hopefully more consistent.