Thanks for the screenshots, it helps. It looks like maybe the pixel scaling is broken in those drivers, pixel scaling is used by default in the GUI (as the images are 1:1 and don't require any scaling) and in the emulator by default.
Press F5 in fullscreen mode to switch to the bilinear filter and it will likely fix the game fullscreen issues.
Someone on another forum had the exact same issue on ATI hardware with one of the earlier versions but I thought it was fixed. I may put an option in which allows you to always have a filter on as it may fix the issues you are seeing if pixel scaling is broken, unless....
Are you forcing mipmapping in the global options? MIPMAP should be set to quality, and if there is an option to "always enable" or "force on" it should be turned off. MIPMAPs are basically textures that are smaller than the original. Why would a driver allow you to show textures that are less size than originals? They are shown when it's appropriate for speed and quality reasons. In RetroCopy there is never a need to have this as I turn off mipmaps manually for textures that shouldn't have it and enable it for those that should.

I believe forcing MIPMAPs is only possible through other apps like ATITools. Another thing to look at is LOD BIAS, if you've been playing with something like ATITools it's likely what is causing the issues.
It looks to me like it's using the half size mipmap version of the GUI textures then scaling it to full size. Either that or the ATI drivers for that series of video cards is broken when using GL_NEAREST. If it's the latter I can offer a fix. If pressing F5 in fullscreen mode increases the quality (take another screenshot) I guess it is the latter issue. Make sure you look at the MIPMAP stuff in your control panel first though to see if that fixes the issue.