Two squares to clear versus three. That is the first difference, and it decides most games at club level before anything about strategy comes into it. Short castling needs only f1 and g1 empty, so a knight to f3 and a bishop to c4 or e2 and you are done by move four. Long castling needs b1, c1 and d1 empty, which means the knight, the bishop and the queen all have to find homes first.
Safety against activity
After O-O the king sits on g1 in the corner, with f2, g2 and h2 untouched in front of it and the rook next door on f1. It is the tidiest arrangement in chess: no gaps, nothing hanging, and the h2 pawn defended by the king itself.
After O-O-O the king stands on c1, one file further from the corner, with a hole on b1 and the pawn on a2 undefended. In exchange the rook lands on d1 immediately, on the file where the action usually is, without spending a move getting there. A short castler often needs Re1 or Rd1 afterwards; a long castler already has the rook where it wants to be.
That is the trade in one sentence. Short buys safety, long buys a tempo and a better rook.
Opposite wings
The interesting case is when the two of you go different ways.
7
2
c
d
h 1 Once the kings are on opposite wings, the pawns in front of your own king are guarding you and the pawns in front of theirs are ammunition. So both sides push. White throws g4, h4, g5 at the black king; Black answers with a6, b5, b4 at the white king. Nobody has to worry about weakening their own shelter, because the attacking pawns are on the far side of the board.
These games are decided by one tempo more often than by one pawn. If you are two moves faster than your opponent, you can afford to give up a piece to get the lines open first. If you are two moves slower, that same sacrifice is a resignation. The way to feel the speed is to solve positions of both kinds: kingside attacks against a castled king on g8, and queenside attacks against a king on c1.
How I choose
Three questions, in this order.
Where are my opponent’s pawns pointing? If Black has played g6 and Bg7, the kingside is solid and hard to attack, so my rooks and my king often go the other way.
Which centre file is going to open? The rook I most want on that file should be the one that does not have to travel.
And how fast do I need to be? If I am behind in development, I castle short and get on with it. Long castling with a loose position invites everything to arrive on c2 and b2 at once.
If none of that is clear yet, castle short. It is the move you will play in most of your games, and it is almost never wrong. There is more on how the two set-ups feed into whole opening plans in castling queenside.