Are there any differences between the original Age of Empires 2 game and the HD remake? If so, what are they?
There are differences between the two games, but they are mainly graphical or Steam-related. As stated on Wikipedia:
The HD Edition includes both the original game and the expansion The Conquerors, as well as updated graphics for high-resolution displays. It also supports user-generated content through the Steam Workshop and multiplayer games provided through the Steam servers.
There are three paid expansions, The Forgotten, The African Kingdoms and Rise of the Rajas which are exclusive to this version of the game.
It's also important to note that a team keeps working on patching the HD version of the game. While it would be ridiculously long to enumerate all those changes here, we can at least say they cover a lot of different points: upgrading performances and rendering, fixing bugs, balancing the units, upgrading the AIs and other convenient points. We can for instance quote this question where we have been talking about the possibility to move a greater amount of units comparing to the original version of the game.