The psychologist/detective choice may make a difference in how the mafia plays.
If there's a psychologist, then they'd be able to rotate who does the killing and they'd have nothing to do late in the game. The psychologist's best odds of finding a killer are tonight, as there's a 2/9ths chance of finding someone, but it'll slowly go down after that. If there's a detective, the detective would have nothing to do on N1 but would have the same 2/9ths chance of finding someone tonight, with the odds of finding someone going up over time as more people die.
So if the mafia lays low for the beginning and cycles through murderers, the psychologist will become pretty useless pretty fast.