sounds to me like the neighbours router is creating the wi-fi network ok, but there is no actual internet connection there, so you can connect to the network, but not get past that to the net itself.
might be worth checking if your neighbour can access ok, if he's paid his bill
etc. might also be a problem connecting to the isp, but if he can connect and you can't, then it could be something a little weirder. if that's all good, a rest of the modem / router is the way to go.