What is the difference between "don't care" and "don't mind"?

Solution 1:

The way I have heard Englishmen say it, "I don't mind" was used only when the person using this expression wanted to express his moderate acceptance of the outcome of a situation. "I don't care", however, can also be used to express indifference and to indicate that the outcome of a situation has no significance for you.

This may sometimes attribute a negative connotation to the message using this expression, whereas—as far as I know—"I don't mind" does not bear a negative connotation. So "I don't mind" is replaceable by "I don't care", but not always the other way round.

Child: "Mum, can I go out and play potentially dangerous ball games like football or soccer with my friends?"

Mother: "I don't mind—the risk is worth the healthy consequences of sport, therefore I take notice and don't object"

OR

Mother: "I don't care—do what you will, your activities do not concern me, I lack authority to form an opinion or I refuse to do so."

Solution 2:

There is a different meaning for don't mind, something like don't object.

But in the use you are talking about, the difference is as you've described. I think of it in terms of what's implied:

I don't mind

means something like "thank you for asking, as I might have had a preference, but actually I don't care to express one. "

I don't care

can mean something like "I haven't a preference, and there was no reason why you should think I might have."

Solution 3:

Mind X means to object to X.
Care about X means to have some interest in and feelings about X.

In the negative, then,

• I don't mind X = 'I have no objections to X'
• I don't care about X = 'I have no interest in or feelings about X'

Both can function as assent to a request for permission

• I'm considering turning on the air conditioning.
  What do you think?
• I don't mind ~ I don't care = 'Sure, go ahead'

Though, of course, they display different emotional attitudes.

Such negative statements seem extremely common in languages of the world for reassurance.
This is similar to the common English expressions
No problem(s), No trouble(s), No worries, Don't (you) worry ((your pretty little head) about it),
and the common Spanish expressions
De nada, No hay problema, No te preocupes (sobre eso), etc.

Solution 4:

While in some contexts it may be no more than a matter of emphasis, there is a substantive difference between the two:

"I don't care" conveys total dismissal of the subject to which it is applied, while "I don't mind" suggests that although it registers, it will not cause distress or disturbance.

edit: Rereading the original post, it seems the quote which prompted the question makes the same point. My contribution is therefore limited to pointing out that it actually is a substantive difference.