Other than battle.net/LAN I don't see any difference.
The "HD" version uses a DirectDraw wrapper which improves compatibility with some systems and provides a menu for configuring the aspect ratio and other graphics options. Other than that, it really doesn't do much. Despite the name, the HD version does not actually render the game in a higher resolution.
If you're looking for the original Diablo with true HD resolutions and several other enhancements, check out the DevolutionX mod: https://github.com/diasurgical/devilutionX