The "looks" issue is certainly a filter issue. As a first step, try to use no filters to get the default look.
When you run in windowed mode, the window is the size dictated by the resolution you request from the game. Lower resolution, smaller window, higher resolution, bigger window. The one thing that doesn't change is that 1 game pixel = 1screen pixel.
When you run in full screen mode, the resolution is "stretched", so it is no longer true that 1 game pixel = 1 screen pixel, instead, 1 game pixel = # of screen pixel/# of game pixels. This means your computer has to convert "on the fly" and I know for me it just crashes the game since my computer sucks. It sounds like yours can handle it, but doing the colours conversion for every pixel multiple times a second is causing you crazy cpu load.
Long story short, what happens when:
1- you run in windowed mode, at native resolution, no filter?
2- you run in full screen, at native resolution, with no filter?
The game should look tiny surrounded by a lot of black. You should then be able to use the battlescape and geoscape scalings (I can't remember the exact term, but it's in the video options. Something like 3x game and 1/3 display) to make the game more visible.