'Take responsibility as a mom', means that you accept the responsibility for something because you're the mother. As a mom, I take responsibility for my children's diet. [Couldn't use take on here] 'Take on the responsibility as a mom;, suggests they will take on a new responsibility. As a mom, I'll take on the responsibility for improving my children's behaviour. So 'take on' suggests you are going to 'start' a new responsibility.

Mari, those both look good! They can mean the same thing. It simply depends on how you want to use them in a sentence. Both can be used in the imperative form, for instance. If I "TAKE ON the responsibility as a mom," it means that I've decided to do what is necessary to be a good mother. The second sentence could be formulated as: "I need to take responsibility as a mom and advocate for my child." Again, they mean the same thing. Just depends on how you want to use them.

"take on" means more "be willing or ready to do something". If someone is going to "take on" the responsibility of being a mom, I would assume they are about to be a mom and are not yet. She is thinking about having a baby or adopting, and she is ready. Example "That's a difficult puzzle to take on..." (The puzzle is a hard challenge to commit to) "I'm going to take on the task!" (I'm going accept and start the task) "take responsibility" like "take this" is a command. I would see this as someone commanding an irresponsible mom to be more responsible. "Take responsibility for your actions!" (Someone probably yelling at someone for being irresponsible)

To me they mean slightly different things, but it's a subtle difference. When you "take on the responsibility" for something, you didn't have that responsibility before. It also sounds formal; the responsibility is well defined, like a position at work. When you "take responsibility as an X", it sounds like you were already X, but you need to act like it. So you were already the mom, but you weren't doing responsible things a mom might do.