"See you later, alligator" as a farewell dates from the 1940s or earlier, and quickly picked up the riposte "In a while, crocodile" (or "After (a) while, crocodile"). The phrase was the title of a hit song by Bill Haley in 1954.
For the purposes of the phrase, there is no difference. The phrase uses rhyming. Two friends are saying good-bye to each other, and each turns what they say into a rhyme, by calling each other an alligagor/crocodile:
(I'll) see you later, alligator.
(I'll see you) after a while, crocodile
"See you later" rhymes with "alligator".
"After a while" rhymes with "crocodile".
This was a 1950s popular song in the US. Those 2 lines (repeated several times) were the entire chorus. They made up a lot of the song. I still remember the melody. Lots of children liked the phrases and repeated them constantly.