Grrr. When it works technology is great, but when it doesn't ... Grrr.
So, I install OS 3.1.3 to keep my phone up to date. I subsequently updated a few apps and bought one other, then after a while I shut down my phone to clear the RAM as I wanted to play a memory hogging game.
After restarting the phone, it stalled at the Apple logo for over 10 minutes so I performed a forced shut down and reboot.
Ironically, as one of the fixes 3.1.3 was meant to introduce was the opposite of this, after it finally rebooted successfully I found that all my third party apps would not launch (they'd briefly flash open then quit straight away). In contrast, the Apple-supplied ones would launch fine and stay open and usable. I tried rebooting a few more times to see if it would fix it. Nope. Then I discovered I couldn't synch my phone to iTunes either. Bugger.
Luckily, before I went the nuclear route of restoring my phone (which is a massive PITA), I discovered the simple fix so I'm posting it here in case anyone else should have this problem (which seems to be surprisingly common if the number of threads at Apple's support forums are anything to go by, and they stretch back a long time, to the days of OS 2.0.x).
The fix:
Download and install a free app*.
Problem solved. You should be able to launch your third party apps again.
It seems that being forced to input your username and password again by an install will fix the issue.
* Or, if an update is available for any of your current apps, download and install the update; this should have the same result.