Well let's talk 8-Ball becasue I assume that's what you are asking about.
APA/CPA
If you make a ball on the break you have to take that group (stripes or solids)
Making the 8 on the break is a win.
If you scratch while shooting the 8-Ball it's loss of game.
You do not need to nominate a ball or pocket until the 8 ball so if you fluke one of your balls you keep shooting.
ACS/BCA
Table open on the break.
If you make the 8-ball on the break you have the option to spot it and keep shooting or just break again.
If you are shooing the 8 and scratch the cue ball it's just ball in hand for your opponent.
You must nominate a ball and pocket on every shot, you may also call just a shot where you can make a ball and force your opponent to shoot after.
Well that's what I can think of for now, I know there is also a difference when balls are knocked off the table or made during a foul. I think in APA/CPA they are put back on the table.
Hope this helps,