Possible new fix
I joined this forum just to post some possible help. This problem is *very* frustrating for those of us who spent quite a bit on this phone.
I've had the same problem mentioned before. All of a sudden, a downloaded app will stop working. The initial screen will come up, and then immediately go back to the home screen.
Once that happens no third party apps will load at all. They will all immediately crash in the same fashion. I've tried the reset/resync/restore and this only worked once. Rebooting (holding down the top button and watching the oh-so-long apple boot up) never fixed the problem.
I deleted all my third party apps, and did a re-sync. None of the applications were copied over. I noticed there was one app showing under "Applications" in iTunes that never synced (learn italian).
I clicked on "Applications" under Library, which showed me a list of apps that should be on the phone, but were not currently. When I double clicked one (twitterific I think) a message popped up saying that I had not yet authorized the use of the app on this computer. I clicked yes or whatever it wanted to authorize.
When re-synced again, I could clearly see the applications syncing in the progress bar, something I had never seen before. After the sync, the apps were downloaded to the phone, including the lost italian app.
All my apps are now back and working.
Hopefully, this helps someone else. Previously I did the full re-sync of my complete iPhone twice before (45+ minutes) and it's not fun. Especially when the problem didn't get solved.