"To brag" is a one-time thing. The person spoke once about his accomplishments. "To go bagging" suggests a continuous action. The person spoke often about his accomplishments.

The first one (You don't have to go bragging about it) emphasizes that the person went around to many different people bragging. The second one (You don't have to brag about it), I would use if I were talking to somebody and they were bragging to me about something.