By a PET scan, you are referring to a Choline PET scan. Choline is the agent used.
PSMA is a different agent, so that's a PSMA PET scan. i.e., they are both PET scans.
PSMA will find spread to nodes whilst they're smaller, which means there's a very slightly better chance of catching them all in the primary treatment, rather than coming back for salvage treatment. It's not a large difference, but something to bare in mind.
There have been problems manufacturing the Choline agent on and off over the last 12 months which has caused long waiting times a few times in that period.
They are both quite a large dose of radiation, compared with something like an X-ray. Choline is the equivalent of 5 years background radiation, and PSMA equivalent to 3-4 years background radiation. You don't need to worry about this unless you intend to have lots of them.
Both require a PSA > 2 to be able to give reliable results, but when PSA is < 2, PSMA gives more reliable results than Choline. (For PSA < 2, a Carbon Acetate PET scan is most reliable, but there are none available in the UK as far as I know, not even private.)
If I recall correctly (and I might not for this), a Choline PET scan should be done before starting hormone therapy (because hormones reduce the activity it's looking for), whereas PSMA is not so fussy about that and attaches to prostate cells even when they aren't rapidly growing. (This has been an issue with the delays for Choline scans where patients have to delay starting hormone therapy too.)
Cost of a PSMA PET scan in UK is around £2500. Sometimes, it can be had on the NHS, but usually not. It can be done cheaper in some other countries if you are prepared to travel. If I recall correctly, BUPA required a PSA > 20 to pay for a PSMA PET scan when I asked (which seems bizzare when it's best at low PSA levels). Maybe they will if a consultant requests one at any PSA level - I don't know.