Off the top of my head, the only thing that springs to mind is that "administer" is maybe more common in the senes of "give (out)". So if you want to avoid connotation with that meaning, "administrate" could serve this purpose (though you could also "manage", "organise", "steer"...).

Just possibly "administrate" suggests more closer being an "administrator" in the sense of it being one's job title/function (as opposed to just administering something on an ad-hoc basis).

So apart from that, it may just depend on whether you like the sound of verbs ending in -ate...

Off the top of my head, the only thing that springs to mind is that administer is maybe more common in the sense of give (out). So if you want to avoid connotation with that meaning, "administrate" could serve this purpose (though you could also manage, organise, steer...).

Just possibly administrate suggests more closer being an administrator in the sense of it being one's job title/function (as opposed to just administering something on an ad-hoc basis).

So apart from that, it may just depend on whether you like the sound of verbs ending in -ate.