OK, either I misunderstood the wording of the question or I am looking at things the wrong way.
From what I understand you have two laptops. One is working perfectly and is connecting via wireless with no issues. The other will connect to the network wirslessly, but no pages will come up.
Am I understanding things correctly?
There are a number of possible causes for this. So many that the simplest thing would be a remote session. I will propose a remote connection and if you want to go this route we will figure out the next set of steps.