My guess? An old world computing center: alpha and beta (and maybe gamma, delta...) are currently running on whatever is left of the old world infrastructure, giving them clear limitations in accessibility and computing power. We have seen multiple times that alpha needed physical access through Akira in order to circumvent security blocks, and now we see that they are competing for computing resources. Having access from a control point may give her access to ALL the leftover net, as well as a considerable boost in computing power. Or maybe they are trying to reconnect with the main AI, of which they are disconnected subroutines...
The other guy is looking for a treasure trove of old world technology, as well as the location of all the other ruins still active.
Or maybe he got a better insight on what Alpha is actually trying to do, and want to rebuild the world WITHOUT the AIs that caused the downfall the first time...