Why is it "King's Road or Queen's Road" but "Princes Road"? Also Wikipedia says that "Kings Road" is okay but would it be alright to say "Prince's Road"?
The apostrophe in names is a matter of spelling. Consider it the 27th letter.
Downers Grove, Illinois used to be Downer's Grove, but in 1873 it lost its apostrophe. Until then, it was spelled with an apostrophe. Since then, it's been spelled without one. This is the sort of fact you need to memorize or look up, not something predictable from first principles.
I suggest you spell Princes Road and King's Road the same way everyone else does. Why? No reason other than consistency—we're used to seeing things spelled a certain way, and departure from spelling conventions only causes confusion.
Originally it seems to have been the Post Office that dropped apostrophes from street and place names (possibly to make automated sorting easier?) and now many councils are making it even more official, as discussed in Kris's links.
The apostrophe brings out the etymology of the name, but the names have lives of their own, independent of the etymology, and their simple use as labels takes priority.