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
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).