Radeon X1*** series cards suffer from driver issues with GLSL (Graphics Library Shader Language) on new Catalyst drivers.
Since shaders are part of the latest mediaVP's, the game automatically attempts to use GLSL, but the bad drivers cause the game to crash.
You can easily test if this is the problem by ticking the "Disable GLSL (shader) support" on the "Troubleshooting" list of features on the Launcher. If the game does not crash when this option is enabled, then this is indeed the problem.
Obviously, this is undesireable because then you lose quite a bit of graphical goodies. The solution to this problem is to use an OpenGL driver DLL from older set of Radeon drivers. To do this:
1. Download
this file.
2. Extract the "atioglxx.dll" from the 7zip archive into your FreeSpace2 directory (for example, C:\Games\FreeSpace2\ )
3. Start the game (obviously, with the GLSL enabled).
This DLL file has a good track record of solving the problem(for XP users; Vista/7 users have to just disable the GLSL AFAIK), but remember that since I haven't actually seen the debug log of your installation yet, I can't be sure this is the exact problem you are having, but it's symptoms match with the previous users with this issue (Radeon X1*** series card, random crashes/freezes) so it's the most likely reason for the issue.
Good luck and let us know how things work out.