The most common and most likely answer to this crossword clue is the 6 plus 3 plus 5 letter word BEFORE AND AFTER
I've seen this crossword clue in Financial Times
B
E
F
O
R
E
A
N
D
A
F
T
E
R
Buddy explains!
The straight part
We know that "See what a difference X makes" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case see what a difference x makes = before and after.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.