What is the difference in English: Onlar bebekle ilgilendiler - they took care of the baby vs Çocuklar ile o ilgilendi - she has taken care of the children. I know that "she took care of the children" is accepted, but is there any difference between "took" and "has taken"?
Yes, a small but subtle difference. "Took" suggests something that happened in the past and doesn't happen anymore. "Has taken" also refers to a past action, but I'm not saying that it won't happen again. For example, "when I went to Istanbul last month, my neighbours took care of my dog." "has taken" would be completely wrong here because I'm talking about an event that is fully in the past. Now, if I want to go back to Istanbul and am looking for people to take care of my dog, I could say, "maybe I can ask Kadir - he's taken care of my dog before."
EdvntL has stated the differences very well. I might add that one can use "has taken care of" to refer to the immediate past, to indicate that something has just recently been accomplished or completed. E.g. "Have the children been looked after? Are they all right?" "Yes, don't worry, Mary has taken care of them. They are fine."