To take no notice of is transitive. You have to specify the object that you are ignoring.
Don't care is intransitive. If you don't care, what you don't care about is implied by the context.
Sometimes it may seem otherwise, as in I don't care what other people think about this. But I think the best grammatical analysis of that sentence is that the preposition about has been omitted. I don't care about what other people think about this.