FIX for when apps won't connect but you can get to sites using Internet browser (Wi-Fi device)
If your e-mail isn't updating, Play Store won't connect and other apps won't work/update, but you ARE connected to Wi-Fi and CAN access websites using the Internet (browser), check the Date on your device (on the Home Screen). If you find the date set to Jan 1, 2012, go into Settings and under the System section, tap Date and Time and set it to the current date. Just as suddenly as it stopped working, the apps will now connect.
I was totally surprised that this fix worked, so wanted to post to help others that run into this problem (since it's such an easy fix and such a confounding problem!).
If you find out why, or how, the date suddenly gets set to 1/1/2012, please let us all know so that we can all maybe prevent this from happening in the future. And... if you find out why the wrong date causes app connections to break, that would just be nice to know because... I don't think I've run into a more bizarre software problem on any computer, EVER.
Re: FIX for when apps won't connect but you can get to sites using Internet browser (Wi-Fi device)
Guess more like another question, my phone works the way you describe but does not have an incorrect date. Is there something else I am missing?