I can't help it - it implies that you have a choice outwardly, but emotionally you can't do it.
"I can't help it - I just don't like peas." There is nothing in the outside world that would make you dislike peas. It is just a feeling you have that you can't overcome.
I can't avoid it. - You don't have a choice. Some outside force is making do or not do something. Your feelings are not involved.
"I can't avoid it. I have to go visit the doctor about this knee."
"It looks like I can't avoid having to take French this semester. I need it to graduate."