There seem to be two fixes for the problem of iPads forgetting wi-fi passwords and leaving off the "forget network" button:
1: Be attached to a network with a real name (ie NOT the default name that came with the router, default/netgear/linksys)
2: Manually enter the network under the "other" option (as described above)
From this I would guess that Apple has installed an undocumented security feature in the iOS for the iPad to keep it from broadcasting your network security codes everytime it gets within range of a router using default settings. It might help keep people from getting your security codes with a packet sniffer, but because it's undocumented it is a pain in the tail.