Is there any advantage to running form FC as opposed to PF?

For Diadem - is it any different running with FC airship as opposed to PF in Foundation?

You can access the Hardmode version with your FC's airship. That's the only thing if I recall.

Since HM only allows Premade, almost everyone in HM are doing it for 210 gear. You don't have to deal with Gatherers or random people leaving mid-run.

If you gather, you can use an FC ship to make an undersized party of 2 or something, sac pull each other to flight and then gather without being anyone's detriment.
For farming gear, FC premades guarantees you get into the same Diadem instance if you form up as an alliance.
Also, a full alliance is a self-contained S-rank farming unit that doesn't rely on outside groups to help. Been in a few HMs recently where normal comp groups (1 tank 2 healer) couldn't get enough damage credit due to the sheer mass of people at the camps, whereas an organized alliance could just set up shop in a quieter spot and get more ideal credit on the trash mobs.