On Windows, Remote Desktop is simply a "remote-control" type of connection; everything you do on the local machine is actually being performed on the remote host. Only screen updates (like the mouse moving and changes to the screen itself) along with any data transfers are sent back and forth.
That being the case, playing games remotely through Remote Desktop is generally a bad idea.
For one, you have the input lag of your controller inputs (keyboard & mouse) being sent to the remote host & vice versa. Most importantly, the screen updates will be nowhere near as fast as the screen updates on your local machine. While this may not matter as much for simple stuff, like a Flash game, it makes a huge difference when you have a graphically-intense game running at 1080p+resolution with tons of colors. Not only will it refresh slowly, you will have messed up colors as it tries to compress them, and everything will seem to lag as Remote Desktop tries to update your changing screen. All of this can even get worse if your Internet connection is slow as well.
So, end of the story, don't try to use Remote Desktop for anything graphically-intense. You're better off upgrading your local machine to something that can handle your game better, or perhaps just getting a new machine altogether.
Edit: Ninja'ed by the above. Ah well...