I have a native resolution of 720p on my budget HP 2000 laptop. Why can I tell the difference between 720p and 1080p? Is it not correct that with a maximum resolution on my monitor of 720p, I should only see up to 720p? But, Only when the video is not fullscreen can I see the difference.
Your screen is actually slightly better then 720p - 720p is 1280 x 720, but your screen is better 1366 x 768 (which is still a far cry from 1080p) This doesn't fully explain the difference, but it does give some clues and possibilities -
You have not advised "what the difference looks like". It is entirely possible that your system is "downscaling" the 1080p signal to more then 720p - in which case you will be able to see it but it will probably look less sharp then it would in 1080p (look out for greys in black text on white background for example). It is possible that you cant see the difference when playing video because of the motion, or it is possible that your player is bypassing the OS and rendering directly to the screen at the correct resolution in full screen mode.
Its also possible that the drivers for your HD6310 are able to provide better accelleration/decoding at 1080p then at 720p. While it does seem somehwat strange to me, looking at the Specs for your GPU platform here I see it only claims support for 1440x900, 1400,1050 and 1024,600 for the internal connection (and higher for everything else). I accept that this is most likely wrong, however the document does come from AMD and it seems strange they would mention both a lower and higher resolution then native, but not your native resolution !