Merriam-Webster should give us an indication, with two different definitions:
care (intransitive): to feel interest or concern . care about freedom
care (transitive): to be concerned about or to the extent of . don't care what they say. doesn't care a damn
In summary, the transitive sense:
Is more restrictive, since "interest" is not part of it. In fact, as the examples show it is used chiefly in the negative, i.e. to deny that one is concerned. Furthermore, there is a limit on the construction: one could say "I don't care what they say", but one wouldn't say "I don't care my friends".
May imply a degree: I don't care (a bit, the least, a damn, a hoot, etc.)
(In the second case, however, I would rather consider "the least" as an adverbial phrase, as it would be in e.g. French, but that might be question onto itself.)
As matter of fact, I have the suspicion that this particular phrase "I don't care what" is an ellipsis of "I don't care about what", but I don't have enough elements to support that assertion, except that it belongs to a more casual register.