Yes, the Church gives us a great deal of freedom as to music.
Generally, all hymns used during Mass should be taken from a book approved by the bishop/archbishop of your area, or by another bishop or council of bishops in your country.
There is an official hierarchy of desirability:
- Gregorian chant (which is supposed to be the default option for all Masses, and to take the principal and most honored place in Catholic church music)
- Polyphony singing
- Organ-accompanied singing
- Everything else.
Both hymns and “praise and worship” stuff is all under “everything else.” No big difference, as far as the Church is concerned. (Assuming the praise and worship song isn’t something unapproved.)
Singing four hymns at Mass is not the default option; it’s a replacement option, for parishes who do not sing a setting of the Introit, Offertorium, and Communio antiphons with psalm verses (yes, they are part of the readings for every Mass, and they change every Sunday just like the readings). Ironically, most Catholics now live and die without ever hearing the “normal” thing.
I’m kind of surprised that “praise and worship music” hasn’t picked up on all the liturgical antiphons and turned them into choruses for praise and worship. For example, this week the Introit antiphon (ie, the one replaced by the opening hymn) is:
“Sing a new song to the Lord! Sing to the Lord, all the earth!
Truth and beauty surround Him.
He lives in holiness and glory.”
(And then the psalm verses - or you can just sing the antiphon.)