They are related but not exactly the same. "To bring us a child" means to raise that child from a young age and to have a lot of influence over them. "My mother brought me up to have respect fro adults." "I was brought up by my grandmother after my parents died." It's not a short term thing. "To look after a child" is to have care of them for a particular moment in time. If I go to the neighbour's house and babysit their kids for the night and then leave when the parents get home, I have looked after the kids (for that night) but I haven't brought them up.

They are similar but not identical in meaning. A person who "brings up" a child is acting as a substitute parent while a person who "looks after" a child may just be babysitting. I think these can be used interchangeably unless the distinction is important. For example, if the child is primarily cared for by his nanny I might choose "brings up" but if the child's parents are actively involved in his care but occasionally leave him with a nanny I would probably choose "looks after".