Is "administrate" a valid English verb? What's the difference between it and "administer"?

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.

"Administrate" appears to have a history going back almost 400 years (rather to my surprise), and is an exact synonym for administer.

The NOAD reports that administrate is a less common term for administer, when it is used to mean "manage and be responsible for the running of a business, organization, etc."

Administrate has been first used in the sixtenth century, and it derives from the Latin administrat (administrated), from the verb administrare.
Administer has been first used in Middle English, and it derives from the Old French word that comes from the Latin administrare.

The word, "administrate" is indeed listed in Merriam-Webster as a verb. However, I would use “administer,” which is the more common/usual word. In the business world, “administrate” often has the connotation of work handled by a secretary, not a professional.