Hi Moe,
No notable difference between any of the 350/360 series. I know the 350 I have has a slightly different intake attachment, the bar cover has an outside adjuster compared to one in front of the saw, and the air filter cover attaches differently, but same general configuration. Not positive how many UT #s with different 350/360 varients, but other than decals and paint, I doubt you'll find big differences. Some variations may have a different carb, but again, I doubt it will be significantly different.
Dan
p.s.
Have you figured out your ignition problems yet?