Brainer There is no meaningful difference. Both phrases refer to engaging in a happy and carefree activity. To do something with "abandon" or "gay abandon" refers to the same pleasant feeling. You could construct an English sentences with other adjectives also which means the same. Example: I began acquiring the gear for my camping trip with Della with the carefree abandon of a youth in love. In abandon, I began buying books on stock market investment. With gay abandon, I rushed down to the common area to see what people were saying about the recent events on the other side of the world. .

Brainer: Maybe it is a good idea NOT to say "with gay abandon." Some people will laugh because nowadays the word "gay" has a meaning that it did not have when this phrase was first used.