It really is down to the latter processing and how good that is. In your case it seems to be RGB vs component so shouldn't be huge. Other may also have the option of interlaced vs progressive (not available on the 3910 from memory). This is one to try but not all will work so you need to find out what the Sharp is capable of.
The best HDMI output I found was Y Pb Pr 4:2:2 (Component) which is what DVD is encoded in. Depends what your Sharp panel does with it.
RGB over HDMI is not quite as good - less detailed, Y Pb Pr 4:4:4 slighty better.
My comparisons were with a Panny S97 and Hitachi PD7500.
Every kit is different though.
i have the 3910, with HDMI i use 576p on with the component option mostly, altho some DVDs i use the DVD player to upscale to 720p or 1080i, i've never found the RGB to be superior yet, but its possible it may work better on some discs.....
however, i set up a component cable last night from 3910-3803amp-7205 PJ, and i actually find the component cable is carrying the picture better than the HDMI.....but this might be because its a long length and i have lots of cables, speakers and computer equipment roaming around the room....its possible my HDMI isnt shielded enough despite being £110......or the PJ isnt so swift with HDMI.....*shrugs*,......the main differnce is blacks, the component seems to handle them better, i get the tiniest bit of black/grey pixellation on some poorer DVD transfers with the HDMI......but the component had no problems getting the full signal there....unless its the 3910's HDMI output is a bit crap....i dunno.....
I found that via HDMI my 3910 wasn't showing blacker than black. You can test this by viewing the black set up in a THX setup found on incredibles, finding nemo etc. You may know this already, but if you are getting BTB you will see a shadow to the THX sign when you turn the brightness up. If you don't then you are not getting BTB. I found I did get it on component, but not on HDMI.
However, when I turned contrast up to 4 on the 3910 it did show the shadow via HDMI. I then readjusted contrast correctly via DVE disc (or you can use THX display) on the 7205.
The result was much better depth of field via HDMI than before. If you don't get BTB I think you miss on finer black detail, which will affect general look of picture.
HDMI cables are VERY high quality generally as a group but some are better than others, the issues you see are more likely down to the difference interfaces / set up.
The DVD disc I thought contained 4:2:0 info? Could be wrong. SDI output 4:2:2 component as this is what comes out of the MPEG2 decoder and my choice is to use HDMI 4:2:2 component 'INTERLACED' output to external video processing for the ultimate, again it is down to where the deinterlacing / scaling is done best.