Jim,
I'll jump in mostly with my opinions. I would also be interested from hearing from anyone who replaced the 2 series with the 3 series on their classic Cornet.
I have the classic Cornet with the 270 transformer. Several of these failed when the Cornet was introduced. Jim then revised the Cornet to use the 370 series. I never had to change to the 370 - mine still works fine.
There are a few small resistor changes as the 370 delivers a little lower B+ and heater voltage. Jim stated that he thought the 370 transformer sounded better when he made the change. The Cornet and Cornet2 circuits are essentially the same but the C2 board is easier to work with and makes adding the CCS much easier as it's designed into the board (versus some tweaking with parts off the board).
The reason the 370 has a little lower voltage than the 270 is that the 270 is designed for 115 volts primary and the 370 is designed for 120 volts. So the 270 delivers a little more voltage when run at 120v which seems to be the norm these days.
The 270 in my opinion is not a very good transformer. It runs hot after awhile and that seems to change the sonics - maybe the internal resistance goes up and drops the voltage (guess). The 300 series are heavier, run cooler, and seem much better made. I use a 372 (300 volt version of the 370) in another project, so that's my reference.
My Opinion: If you've got the time, you're into soldering and tweaking and you're an Audioasylum type of fellow with a Steve Jobs mentality (it's not right unless it's perfect) - replacing the 270 with the 370 is a good move. It's stable (sound won't change over the listening session as the transformer heats up), and Jim says it sounds better. If you're happy with your current Cornet (as I am) and you don't have much time, pass for now.
The C2 board is bigger than the classic board, so going C2 means starting at square 1 and building your own.
Final point - if you have the classic, does it have the CCS? I built my Cornet with and without the CCS. The CCS is a definite addition. You'll see two transistors on the board if you have the CCS (flat, 3 legs).
Grant