The differences are found here:
http://www.gamefaqs.com/n64/913960-mega-man-64/faqs/33895
The biggest differences that would matter (aside from any possible glitch changes that are not mentioned, if there are any) are the change in graphics (N64 has shorter draw distance, but anti-aliasing makes it more pleasing to look at) and audio quality (N64 has extra compression to fit it on the cart).
I personally think that the PSX version would be preferable (and some people expressed this in more detail above) since it was the original. The N64 version did not change too much, and it probably should not be used unless a good reason can be found for using it, other than practically zero load time (I don't think the load time in the PSX version was that long anyway).
Apparently somebody from Japan made a TAS of Rockman Dash for the PSX (which is the Japanese version of the game), but nobody asked any additional questions about it, so it became kind of obscure. The videos add up to an hour and three minutes, so I'm guessing the complete TAS is in the 50-60 minute range. I have not watched these videos yet, but now that I found them, I might.
I would like to see a TAS of this game too, whether using the PSX or N64 versions. I think it has potential for entertainment. The same applies for Mega Man Legends 2.
EDIT: Just watched the Rockman Dash TAS. Ingame time was 0:46:20. It's pretty well done, with nice humor added when waiting is necessary. There were some odd things though, including a few times where there was damage taken for no apparent reason (minus the few times it was done for humor).