What usually happens at a reading group is that members decide beforehand which book they're all going to read, and then they meet and discuss it, and decide on the next book. So yes, it's just another name for a book group.
Your role as facilitator would be varied, according to what sort of group it is. You could be asked to arrange a meeting place, lead the discussion, obtain and distribute the books (our library has a reading groups' facility, where registered groups can borrow sets of books) and a host of other stuff. It's mainly administrative.
There may be specialist groups that help members to improve their reading skills, and I would imagine your training will cover that if it's necessary. It may also give you some help on knowing how to lead discussions and what to pick out of the books as cues for discussion.