Firstly, you've got the tablet's names all mixed up; the old one is the 770, the newer ones are N800 and N810. (Nokia added the N800 to the N-series, before that it was just numbers.)

The app manager and such are pretty much the same. The 770's last "official" build was OS2006; since then, there have been OS2007HE and OS2008HE, which are backports of OS2007/8 to the 770. They have some components in common with OS2006, though, where the corresponding newer stuff is N8x0 only.

I don't have a 770, so I can't be horribly much help, but I think basically everyone runs one of the HEs; some prefer 07HE, and some go for 08HE.