Does Non-HD Footage Look Worse on HDTVs than on Analog TVs?

I received a great question in my email this morning:

“I've heard several times that non-HDTV content looks worse on HDTVs. Then I read something that implied that the non-HDTV content looks worse on 1080p than 720p and if you will watch much non-HDTV then you should get a 720p set. Is there any truth to that?” - Will

This is something I think a lot of people need to be aware of because I hear so many stories about how people are disappointed when they turn on their new state-of-the-art HDTV only to see an inferior picture as to what they saw in the store.