iPhone 5s won't update apps

My iPhones 5s will not update apps. Whenever I tap 'Update' next to both the apps that need updating, it does a spin thing for a second, and then goes back to saying 'Update.' I've tried closing all apps, signing out of iTunes, and then restarting my phone, but the problem remains, and now I can't even log back into iTunes (I get the message: Cannot connect to iTunes Store).

Unlike other people, it did not ask me to enter my password whenever I tried to update. Even now that I'm signed out of iTunes, when I try to update it doesn't ask me for a password.

My last resort is making a backup of my phone, wiping it, and then restoring from the backup, but I'd like to avoid that if possible.

(Side note: My software is up to date with iOS 9.0.2)