New York City - American Airlines Admirals Club lounge VS Delta Sky Club lounge (JFK) which is better

As of above

It may be irrelevant, you have to pass security to get to them and for that you need an appropriate boarding pass. They are in totally different terminals.

In one location where both clubs are side-by-side (RDU), I tried them both and thought the Delta club had better munchies and was a nicer place.

If the OP is flying both airlines and can access either one, I'd take the SC. Better complimentary food selection. IIRC the DL SC also has showers if that is important to the OP as well as the AC. Of course, both have free wi fi and complimentary wine, beer, and house drinks.

The JFK AAdmirals Clubs have showers, too.

Both sets are nice and I wouldn't be too annoyed in either. Nothing particularly amazing about either to make them more desirable IMO.