How does censorship in the English release affect the Dragon Ball Z manga storyline?
I think you're asking, what was cut in the English anime shows from the original story. I don't know everything, but I do know what I've seen.

Dragonball-had multiple parts cut which had Bulma either getting her top flashed(krillin vs invisible man), I believe a shower scene as well, her privates smacked(early on goku fun), and some language(not a string of curses, just an occasional bad word). Some of the dialog was altered to make it less perverted or more American/kid friendly

Dragonballz had more language cut. Like dragonball, certain phrases are altered to make it more kid friendly.

Editing-it won't ruin anything, tho some parts would be very cut thru. Watch the Japanese version, with English subtitles. Most dvds let you watch the English version with subtitles for the Japanese version, to see what's actually said.