Of course, when connected to an computer, you'll always want 1:1 pixel mapping. Hence, there's a setting in nearly all LCD TV's for 1:1 pixel mapping. I bet that your LG is not an exception in this aspect. The option could be named differently depending on the manufacturer. It could be called "full picture" or something similar. Do you have it set accordingly?
What confuses things a bit more, is that because of the aforementionet default overscan, the fglrx driver underscans by default (to get a usable display in all cases; e.g. menu bars could get lost because of the overscan). In your case, it could also be that the display is already doing 1:1 mapping, but the driver is still causing the underscan (which it does by default) trough the HDMI. But don't give up, there's a fix - search for "black borders" on this forum, there has been a lot of talk about this in the past.