So both laptops connect to the network fine, but only one can access the internet? Can the one that cannot access the internet see anything else on the network? Like shared media files or hard drives?
Have you tried different browsers? Which ones have you tried?
Does your browser come up with a message about
proxy servers? If so, follow the directions in the link.
Is there a way on a Mac to see if the hardware is having troubles? Like Windows device manager.