Running on both WiFi and mobile connection, the browser can connect to the internet ok but apps can not connect.
What can cause this?
Samsung Galaxy S
Sometimes the culprit is much easier to solve: If accessing secured websites (https:) via the browser fails as well, make sure your date/time settings are correct -- otherwise certificates will fail. Best explained using an example:
Say, today the date is May 28th 2013 -- but your device's date/time is set to January 1st 2010. Now you visit a secure website, which is using a certificate issued on June 1st 2012 -- valid vor 2 years. Your browser won't accept that certificate, as it must assume it's a fake -- issued in the future. Same would apply if your browser's date were set too far in the future, e.g. December 31st 2014: the certificate would look "expired" to the browser, which again would not accept it.
Same applies to your apps, which often use a secured connection (hopefully!). Just unsecure connections (not using certificates) would work fine -- which explains why your browsers work on many sites.
This seems to be a software problem. Try a hard reset - that should restore connectivity. Note that this will delete all your data, so back it up first. When installing apps back try to check connectivity after each new installation - in this way you'll be able to determine what causes the problem.
Also as a random guess - do you use a proxy that only permits web-pages (only HTTP protocol)? That would explain why your browser works, but apps do not (they often use other protocols than HTTP).
Disabling the 'data connection' and then enabling it solved my problem. You should try it. You can alternatively try toggling the airplane mode (or use a widget) that should also enable and disable the data connection (both WiFi/data).