Hi Steve and welcome, I've got a 60D as a backup to my 5D3 and I recently bought a 650D for the wife. TBH the 650D probably edges it on technology, better ISO, very good AF (specially for video), good fps rate, Digic5 processor and a great responsive touchscreen.
The 60D edges it on design and ergonomics though, for me anyway, as it has the top screen and second dial on the rear which makes it very easy for altering aperture in manual etc.
As you are upgrading from a 350D, which has a similar body to the 650D, you might be better off getting the 650D as you know the layout already. It has almost the same 18Mp sensor as the 7D, 600D and 550D but the 650D has the hybrid AF sensor which is obviously newer, and the new STM lenses work excellently with the 650D too and there's bound to be more coming out.