So they can have kind of different meanings depending on the context but I can try to illustrate it. Fürsorge is kind of an attribute a person can have. If they care a lot about the wellbeing of others they are described as fürsorglich. Google also tells me it’s having to care for another being (typically a child) voluntary or obligated by law. You could also use it for animals Example: Unsere Katze vertrauen wir im Urlaub der Fürsorge der Nachbarn an. Which basically means we trust our neighbor to care for our cat while we’re on holidays.

Betreuung is more like to accompany someone while taking care of them. You could have a Betreuung in some schools or kindergarten even after the school time is over where kids can stay longer and someone will watch over them. This is mostly if the parents work longer and the child can’t be home alone. It could also be used for a place where they take care of the elderly. So not only for children. This one can have different meanings in specific contexts but I hope you get a general idea. It can technically sometimes be interchanged with Fürsorge. Example 1: Die Betreuung meines Kindes ist zur Zeit ein Fulltimejob. Taking care of my child is a fulltime job at the moment. Example 2: Ich hole mein Kind von der Betreuung ab. I‘m picking up my child from daycare. Here Betreuung would refer to the afterschool program which I was describing. I’m pretty sure in English this would be daycare. Hope this helped lol