Public:
You can join in after the match has already started
you can quit without any penalty (I believe)
maps are chosen randomly
don't ever have to leave the game
Ranked:
You will always start a brand new match
you can pick your loadout/character at the beginning of each match
you can vote on which map you like
quitting will cause you to lose a decent sum of XP
at the end of each match you are booted back out to the main menu and have to search for a new match