You need to understand the difference between: 1. Being connected to your local area network, and, 2. Having a connection to the internet.
This is something that many people don't understand--a lot of people just see "Connected" and think that means that you have a connection to the internet, but in many cases it does not.
What is your set up like? Modem --> Router --> Computer? If so, and you see "Connected", that means that you are connected to the router--nothing more. If that is the correct scenario and you are having trouble accessing the internet, then try to connect your computer strait to the modem. If you are then able to get online, then the problem exists with the router; if you are not able to get online when directly connected, then the problem exists with the modem/ISP.