Your monitor is too big. I payed $800 for my NEC 22 inch 1600x1200 in 2001. Most people were running 15 and 17 inch displays. Homm3 still has extremely crisp hand drawn graphics with good detail, it's just most people are running 20+ inch monitors with 1920x1080 resolutions, and you're either playing with very small stuff, or blowing it up a lot.
It can't handle being zoomed in at all without losing it's fidelity, but at the proper resolution it's hard to beat with each pixel being the color it's supposed to be. You can't get that from a low poly 3d model, it's all straight lines and jagged edges with texture mapping that stretches and bunches as the model animates.
3D brings advantages in perspective, different viewing angles, less expensive animations because you only have to make it once instead of dozens or hundreds of times. It doesn't bring image quality, if you can model it, you can draw it just as well or better from a fixed position. Moving your viewing angle around so you can find resources is a negative aspect of the newer 3d maps, there's no reason to change perspective without them.
I run HoMM3 HD at half my monitor resolution, and when my eyes are tired I blow it up to fill the screen. It's a good size for modern displays at that point, but it's not nearly as defined.
