Any app that did not come pre-installed on the phone will not open. Mail apps, music, OneNote, Office & Xbox hubs, etc. all open fine. Any app I've installed from the store will not open, such as WP Central, Weave, Bing apps, Nokia map apps, etc.
Restart and soft reset did not help so I'll have to try a hard reset, which I won't be able to do for a few days.