First of all, the HD mod is a must. High-res, different renderer, bugfixes, neat new features.

As for mods as in content changes, the Horn of the Abyss mod is hands-down the best thing out there. The developers aim to make it feel like something New World Computing would've developed, and it shows. They've rebalanced the existing towns, added a tenth town (the Cove), new creatures, new map objects, and even new campaigns. The HD mod is not only designed to work with it, it can also install it.

You need the HD mod to get crisp, non stretched graphics on a modern monitor!
The difference is quite dramatic.

Either my laptop screen is not modern (possible, I suppose, since it is a number of years old), or I just don't even register the black placeholder bars on each side (which happens, and completely doesn't bother me).

I intellectually understand that some people have a large monitor with a different ratio than of old, and they want the whole thing used, but I don't really get it emotionally.

Well, after some testing with making screenshots, it seems the results may depend on your graphics card as well.
On my system vanilla HoMM 3 has stretched, blurry graphics on a widescreen monitor.
With the HD mod I get crisp graphics and black borders.
But taking screenshots with IrfanView or Fraps does not show any stretching.

Scale with aspect on GPU. Some monitors also have the option of switching scaling modes.

The basic HoMM III was nice but without flavour.
In the wake of gods is delivering tons of new contents and the best is you have an
option menu to turn on and off the modifications you want.
One thing I really enjoyed was the auto battle option so you dont have to go through
every insignificant fight.Yay!

https://www.moddb.com/mods/h3wog/downloads/in-the-wake-of-gods-358f

The succession wars
Without it the game is ugly af.