Hello!
Realistically Psychiatrists get very little psychology training. They are MDs who take a couple extra courses and do an internship as a psychiatrist. Many states are seriously considering allowing Psychologists to write scripts, which is the bulk of what many Psychiatrists do.

Forensic isnt like TV. The FBI might actually employ a few, but they are more likely to be I/O psychologists and have nothing to do with tracking down criminals. Forensic psychologists might get called in for special cases as consultants. The FBI however has rather few on staff and most of them have law enforcement backgrounds. Forensic psychologists and psychiatrists spend their days preparing for or in court or working in the prison system or similar functions. Being either requires a good bit of hustle unless you just want to take a low pay state job. You have to hustle up clients and build up a reputation for your court manners that make lawyers seek you out for or against clients.

Psychiatrist will make a more money in terms of what they can and do charge per hour. However ones hustle is the real determination in either field. The psychologist is going to have more career options and opportunities. More interaction with people as well.

Hopefully this helps.
Best wishes!