Not quite the same – one is about where your wage is coming from, the other about your education and these days, when we have hundreds of actors playing in many series without any qualifications, the difference is more pronounced, I think. Bracia Mroczek, for example, are „z zawodu aktorami” but they are not „zawodowymi aktorami”. ;)
Nevertheless, there is overlap of meanings and this is a nuance way beyond the level of this course(I would guess it's something you should know about at C1 level and be able to use properly at C2 only), so it's just a side note.
That is a different thing, one is „aktor filmowy” the other „aktor teatralny”, both can be „zawodowi aktorzy” or they can be „amatorzy” and either way, if they earn their wage acting, they are „z zawodu aktorzy”.
The difference is about qualifications, someone who finished formal education(‘Szkoła aktorska” – acting/drama school) has „fach” and therefore is „zawodowy aktor” and isn't „amator”(amateur) and if he works as actor, he is „z zawodu aktor”, because that's what he does for the living…
The problem starts when someone has „fach w ręku”(doing his job really well due to experience gained or talent) and therefore is considered professional even if he doesn't have formal education – in such a case many people would still be inclined to call such a person „zawodowy aktor”, which would make it a synonym for „z zawodu aktor” – as I said, it's pretty hard to explain and the difference is a nuance and not so clearly delineated.