HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

Well, for example, I can't play version 3 because it starts lagging so much that it becomes unpleasant to play. 2.98, on the other hand, works perfectly.

Could you elaborate?
I can understand such a statement if HD 3 uses a 32-bit MMX renderer and one of the bilinear sharper filters.
This sharper filter puts additional load on the processor compared to a simple bilinear filter (in 2.98, 32-bit MMX only uses a simple bilinear filter), and this is only when stretching the game's image.
There's nothing else that should make it slower than in 2.98.

To speed things up, you can try:
1. Don't stretch the window in MMX mode (if playing in fullscreen, choose the desktop resolution).
2. Change the renderer.
3. Disable the CpuPatch option.

Added 9 minutes later

Does anyone know if the map is saved in the host's random maps folder (or somewhere else) now, or is it like in WoG?

The map is saved, but if you're playing with protected saves, it's encrypted, so you can only open it after the host receives the opponent's password.

I'd like to see a list of differences between 3.35 and, for example, 2.98.

If I were to describe the list in detail, it would be very long.
Maybe I'll do it someday.

In general, there are
very informative screenshots
and a detailed list of changes compared to the original
on the website.