There is no real difference in the addressing. I believe there are four address ranges that are used for internal networks and are never used for public IP addresses. 10.0.xxx.xxx and 192.168.xxx.xxx are two of them. I believe another one begins with 169 which usually gets used for "self-assigned" IP addresses (when a network card can't find a DHCP server).
I use the older LinkSys WET11 802.11b bridge with my Airport Base Station for the game systems (XBOX, PS2, and GameCube). I changed WET11 to work on the 10.0.1.xxx range.
I haven't used the LinkSys game adapters, but I would imagine they should be able to work.
Some of what I found with the WET11 bridge from LinkSys may also apply to the game adapters.
With the WET11, I was able to configure it completely from the Mac also. Since they come from LinkSys with the IP address preset to 192.168.1.225, it wasn't possible for me to connect to it from the Mac which was on the 10.0.1.xxx network. I disconnected the Mac from the network, connected it directly to the WET11, and manually set the Mac's IP address to 192.168.1.5 (setting the router/gateway IP address to 192.168.1.1). I could then get to the LinkSys config screen by going to 192.168.1.225.
Make sure you change your Mac settings back to the way they were when you re-connect to the network.