You say GW won't run on the laptop... do you get some kind of error message?
Your router is definitely not the problem, neither is your ISP. The fact that it works fine on your desktop proves that already.
Try these two things:
1 - (Startup Items) Click START, then RUN. Type "msconfig" (without quotes) then hit enter. Click the STARTUP tab on the top. Either disable everything, or disable everything except anything having to do with your wireless (if you don't use Windows to manage wireless). Click APPLY then OK. It will ask you to restart... restart your laptop. A message will pop up when you get back to Windows basically letting you know that the startup items were changed. Check the box to not "show me this again" then click ok. Try running GW. If it runs now, there was something running that was blocking it. If you notice you're missing something after changing stuff here, go back to "msconfig" and start checking boxes one at a time and rebooting until you get it back. The object here is to have as FEW things as possible starting up. But also to find the culprit of your issues with GW.
2 - (hosts file entries) Check your "hosts" file. This is located in Windows/System32/drivers/etc/ (it has no extension. Open it in Notepad). First, see if there are any entries listed in there. This will amount to IP addresses and DNS names (i.e: 74.125.95.103
www.google.com). If you see entries in here (other than the example entries) make a copy of it inside the same folder, and name it "hosts-original" or something like that. Edit the one still named "hosts" from notepad. Remove all entries except the top section. This is all you should see in the hosts file:
Code:
# Copyright (c) 1993-1999 Microsoft Corp.
#
# This is a sample HOSTS file used by Microsoft TCP/IP for Windows.
#
# This file contains the mappings of IP addresses to host names. Each
# entry should be kept on an individual line. The IP address should
# be placed in the first column followed by the corresponding host name.
# The IP address and the host name should be separated by at least one
# space.
#
# Additionally, comments (such as these) may be inserted on individual
# lines or following the machine name denoted by a '#' symbol.
#
# For example:
#
# 102.54.94.97 rhino.acme.com # source server
# 38.25.63.10 x.acme.com # x client host
127.0.0.1 localhost
Leave the "127.0.0.1 localhost" entry. Save the file, then reboot and try GW.
If GW still doesn't run after doing either of those, I'm sorry I couldn't offer more input. I'm willing to bet it's something in your startup though. A lot of people have so much crap loading at startup that is unnecessary, and don't even realize it. I doubt it's your hosts file, but it's possible. Do the first one first, see if that solves your problem.