With today's digital TV signal, I don't think there will be a "picture quality" difference. The main difference, I think, is the ability to capture a weak signal. The larger dish will capture a signal when the smaller dome (which actually has a small "dish" inside) will not.
Once you acquire a signal, the picture quality should be the same as long as you are able to maintain signal capture. In less-than-ideal circumstances (heavy rain, snow, foliage cover, etc.) the larger dish will do a better job of capturing and holding a signal.
We use a portable dome-style satellite antenna (King VQ4550) that was purchased for use until we could get a dish installed on the roof of the coach. We did not like the Winegard units that Newmar installs, so were going to go with a RF Mogul Eagle unit. Fast-forward and we are still using the portable dome. It has provided excellent service to the point where I don't think we will move forward with a roof-mounted dish.
TJ