there is a discrepancy with the English here. The sentences should be unlinked.

"ich mag seine Sprechweise nicht" => I don't like the way he talks.

Well, there is a clear difference between "I don't like the way he talks" and "I don't care for the way he talks", which means it is indifferent to me...

yes, so if I don't care, that means I don't have interest.
But if I don't like (mag nicht), i have a negative interest, not an absence of interest. That is quite different. Actually, it is precisely the opposite...

You're perception is misconceived. "I don't care" can't mean "I don't like".

I don't care for... = I don't like... (and maybe, to a small extent, = I don't care about)

What arcticmonkey says is true, IMO.

>and maybe, to a small extent, = I don't care about

Yes, depends on the extent...I understand that when it is me, it is being reduced...whatever!

I don't care for him. Er ist mir unsympatisch.
I don't care about him- Er ist mir gleichgueltig.

I don't care for your tone of voice= I am displeased by your tone of voice.