Video cards do affect the overall video quality output, and yes in general, more powerful cards can perform better than less powerful cards due to functions that may be limited or disabled on less powerful cards. There is a point of dimishing return though or no return at all. For example. I would guess that a HD 5770 will offer no better playback quality than the HD 5670 or HD 5650.
One thing to remember is at a certain point "quality" becomes subjective as to what actually looks better.
Below is a link to video quality of some video cards that most people would put in a HTPC. The cards that scored the highest are the HD 46xx series and the 9600GT:http://www.xbitlabs.com/articles/video/display/media-playback_6.html#sect0