They do differ, both for video and audio (more for video, though).
Avia has some bugs with the repeat function when displaying test patterns (meaning it won't reliably hold on a single pattern) and the gray scale calibrations are inaccurate. It's easier to do brightness adjustments with Avia, though, if only because its black bars move, making them easier to see. (The black bars on DVE are stationary.)
DVE is definitely better for widescreen displays, but I was very disappointed to find it doesn't include window patterns across the entire grayscale at 10 IRE steps. (It only has 20, 40, 80 and 100 IRE windows, plus a 60 IRE flat field.) DVE does give better tips for making certain adjustments, particularly contrast. Also, the video demo material is excellent. It could be easier to navigate through its menus, however.
I was hoping that DVE would be the one disc that does it all, but it's not. I still carry around the full slate of test discs - VE, Avia, S&V, and now DVE. An HD-capable video generator will be an imminent purchase...