If you lose a home command in MP, its just a lot of AIP you've dropped on everyone else, and you get tossed to another home as a home core.
So you end up with a new home base? There must be some limitation or you'd never lose.
Yes, there are limitations. Once all home command stations (not counting home command cores) of a faction are lost, that side loses. So it doesn't matter if you still have a core, if there are no home stations proper, game over for the human team.
For the home command core itself, it is an odd beast.
Like the home command station, it is a full fledged builder, however
Compared to a home command stations:
Advantages:
-mobililty
-better resource production
-can stack with a full command station, as it isn't actually a command station itself
Disadvantages:
-less durable (IIRC)
-does not provide supply
-does not get unlocked higher mark command station foldouts
Also, like the home command station, the core has an AIP cost on death. It isn't quite as much as the command station loss, but it still is noticeable. This, combined with worse durability, makes it a bad idea to try to use it as a mobile builder on the offense.
And if you lose your home planet, you lose those resources (until you rebuild a normal command station on there), which is often enough to overtake the extra resources produced by the core. This also really slows down players joining in mid-game, as they get a core instead of a full home planet.
Finally, a mobile core only grants 1x to the cap limit of your ships, regardless of how many home planets you have before hand. Keep this in mind if you take multiple homes, as if any of them die, you only get 1 core back, and thus are bumped down to 1x ship caps.
So yea, the home core is there to give players that were taken out some way to still contribute, but it is overall a big setback compared to still having your home station.