What exactly do they mean?
I couldn't care less= I don't care.
I could care less= I don't care.
Which of the two is better?

What exactly do they mean?
I couldn't care less= I don't care.
I could care less= I don't care.
Which of the two is better?
They may revoke my language pedant credentials, but I couldn't care less that some people say "I could care less."
That the original phrase was I couldn't care less+ is well-established; however, +I could care less+ has also been around a long time, is very widespread, and (as the link points out) employs a different intonation pattern--one that to my ear, at least, makes the phrase more emphatic and dismissive than the variation with +couldn't.
If we demanded all our idioms make grammatical sense, we'd have to chuck half of them out. So, I say let folks use whichever suits their fancy, and that people who care so much about the missing negative should care less!
I only know "I could care less" as an American idiom and it doesn't make sense to me, but as zashibis points out, it doesn't matter.
It has a sort of Jewish-American sound to me, which is also suggested in the link

I always think "I could care less" as having more of a lazy and or uneducated sound to it. As a Jewish-American (well my family is Jewish, I don't believe in it), I am aware that "I could care less" means that there is room for one to care, thus not expressing the desired sentiment.
Having grown up in New York I've heard both plenty--there are so many things to be disdainful of.
Both "I could care less," and "same difference" (for "same thing," also disdainful) used to bug me. These days, I let them slide. I've always felt there was an implied "As if" at the beginning of the sentence, and it makes sense that way. For example, someone might turn an "I shouldn't tell him his business" into an "[As if] I should tell him his business."
CK