Is your router listed in the prefered networks?
I had a very similar problem when i first started using the Airport on my iBook (getting neighbour's networks. Can't exactly remember how I fixed it. The prefs for this is a maze, very unintuitive.
The only other thing I can think of it that I've set up locations rather than using Automatic. I did this mostly because I take my iBook into Uni, and I need a slightly different setup there (vpn, web proxy etc.). For some reason something at the back of my mind is nagging me that this somehow fixed my problems.
Perhaps you could try setting up a new location (New Location in the Location drop-down menu on the Networking Prefs page) I have one called home, that I set up as described above. Once set up ensure that it is selceted in the Apple->location menu
The other thing is what security type are you using?