Golf tournaments are a mixed bag. Some are professional only, some are amateur only, and some are a mix of professional and amateur. The important thing is that only professionals can win money. Even if an amateur wins a mixed tournament, they get only a trophy. The money goes to the top finishing professionals.
Haven't checked the rule book for a few years, but the last time that I checked, a golf professional is anyone who plays (or has played) for money. I doubt that has changed in the interim.
The question is not whether there are prizes (of course there are), but whether Tiger Woods could come up and claim those prizes if he suddenly decided to.
If not, what is exactly the mechanism that bars good/professional players from some prizes ?
Technically he could if the tournament organiser let him, it isn't against the rules. The mechanism is that he'd probably be given a really low handicap e.g. a positive handicap so shots are added to his score rather than taken away as normal, with the intended effect that he's no more likely to win than anyone else.