"Six of one, half-dozen of the other"
I think I like the explanation the best.
"Six of one, half-dozen of the other" says that two things which people refer to differently are actually the same thing. A "dozen" is a counting word that represents twelve (12) of some particular item, so a "half-dozen" is equal to six (6) of that item, and saying "six of one" is equal to saying "a half-dozen of the other." The "one" and the "other" refer to the two things which you are saying are not so different. Example: "I say she's a stewardess. She says she's a flight attendant. It's six of one, a half-dozen of the other." Although something has been said in two different ways, they ultimately mean the same thing.