Saying the DR is basically a copy of a modded JCM800 is like saying a Porsche 911 is just a copy of a modded VW Beetle
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It's the amp circuit design that makes by far the biggest difference in sound, more than all the parts put together. Transformers, tubes, cap types, resistor types etc (in about that order) just tweak the fine nuances or 'texture' of the sound. Layout can make a big difference too, especially in higher gain PCB amps.
Actually, the DR is based on (not a copy of - there's a lot that's very different) the Soldano SLO, which is in turn based on a modded JCM800 (or more likely JMP, going by when Mike Soldano was first doing them) 2203, which is a modded JMP Super lead, which is an evolved JTM45, which is a copy of a Fender '59 Bassman 5F6-A! But that doesn't mean the DR is just a Bassman copy... (which I think is itself based on common 50s circuitry familiar to Leo Fender, but I don't know exactly what).
There is a distinct family lineage through all of these amps though, mostly centered around the cathode-follower tone stack stage, which gives them that rich 'Marshally' crunch tone that's almost impossible to really nail without it. But it's the rest of the circuit that separates them in gain and tone.
For example, the main reason the JTM45 - even through the same speakers - doesn't sound the same as the Bassman is not the tubes (which were the same, originally) or even the output transformer, but the fact that the Marshall has about three times as much negative feedback in the power stage, probably by mistake. (They copied the circuit exactly, but forgot to compensate for the different output impedance.) This makes it much tighter and more 'British'-sounding. Changing even just a couple of resistor and cap values can also have a big effect on the frequency response, which in turn will affect the way the amp distorts as well as the EQ itself. There are far larger changes between almost of the amps in the family tree than just that, too... although where there *aren't* it can be just as revealing - for example, some very unusual preamp resistor values which prove that the DR is at least partly based on the SLO, not a separate evolution from the Marshall.
Changing the OT for one of the same basic spec (turns ratio, core mass) will not change any of that at all, hence it can't do more than subtly change the texture of the tone. Different construction (wire type, winding pattern etc) does make a small difference, yes. But not even as much as using a different power tube type, eg 6L6 (beam tetrode) vs. EL34 (pentode), which have different performance characteristics. It's probably in the same scale as changing the brand of tubes (assuming they're *correctly* to the tube spec, which not all modern ones are) - as you know, you can't make a Mesa sound like a Marshall by putting in different 6L6s (or even KT66s), even though the difference in tone can be quite obvious.
... at least to obsessive listeners like us