Wifi comes in several different flavours.
802.11a, b, g and n.
A isn't much used these days. b and g are the most common, and share the 2.4GHz spectrum, channels 1- 13 in the UK. Your built-in wifi card is almost certainly b & g.
802.11n is the new kid on the block, using a different frequency range. If the new hub is 802.11n, you will need a new 802.11n card to connect to it.
To prevent exactly the sort of misuse of the neighbour's wifi that you mention, various mechanisms can be employed:
1) only allow specific MAC addresses to connect to the wifi network (every network card, wired and wireless, has a globally unique address hard-coded). The router needs to be manually configured with the MAC addresses of all wifi network cards that it is to allow.
2) entirely separate from that is encryption of the data transferred between the PC and the hub - WEP is useless, WPA is a lot better, and WPA2 is even better. The PC needs to have the same pre-shared key (PSK) that was used on the wifi hub to be able to decrypt the network traffic.
In summary, you need to ensure that you have the name (SSID) of the wifi network, a compatible physical network card (802.11b, g, n), the same encryption key, and are on the MAC address "white list".
Cadging off your neighbour is bad form (and may also be illegal).
SD