What's the difference?

This is the correct answer but everyone is going all 3000 IQ about English. Yes, English is nonsense, but in this case it’s pretty cut and dry.

goes to look on etymonline

It looks like both forms of "minister" have a Latin root:

https://www.etymonline.com/search?q=minister

minister(n.)

c. 1300, "man consecrated to service in the Christian Church, an ecclesiastic;" also "an agent acting for a superior, one who acts upon the authority of another," from Old French menistre "servant, valet, member of a household staff, administrator, musician, minstrel" (12c.) and directly from Latin minister (genitive ministri) "inferior, servant, priest's assistant" (in Medieval Latin, "priest"), from minus, minor "less," hence "subordinate" (from PIE root *mei- (2) "small") + comparative suffix *-teros. Formed on the model of magister (see master (n.)).

minister(v.)

early 14c., ministren, "to perform religious rites, provide religious services;" mid-14c., "to serve (food or drink);" late 14c. "render service, aid, or medicine; furnish means of relief or remedy" from Old French menistrer "to serve, be of service, administer, attend, wait on," and directly from Latin ministrare "to serve, attend, wait upon," from minister "inferior, servant, priest's assistant" (see minister (n.)). Related: Ministered; ministering.

And for "administrator":

https://www.etymonline.com/word/administrator

administrator(n.)

"one who has been given authority to manage," mid-15c., administratour, from Old French administrateur or directly from Latin administrator "a manager, conductor," agent noun from past-participle stem of administrare "to manage, control, superintend" (see administer). The estate sense is earliest. For ending, see -er.

And "administer":

administer(v.)

late 14c., aministren, later administren, "to manage as a steward, control or regulate on behalf of others," from Old French aministrer "help, aid, be of service to" (12c., Modern French administrer), and directly from Latin administrare "to help, assist; manage, control, guide, superintend; rule, direct," from ad "to" (see ad-) + ministrare "to serve, attend, wait upon," from minister "inferior, servant, priest's assistant" (see minister (n.)).

So while it was an interesting thought, it looks like all four words, "minister" (noun), "minister" (verb), "administrator" (noun), and "administrate" (verb) have Latin roots.