A scrub brush is certainly the most versatile tool for cleaning a bathroom. It's the best for removing soap scum and tough spots, so you'll need it to give your bathtub a good cleaning. And unless you rinse your sink every time after brushing your teeth, you'll probably need it to get the dried toothpaste and soap scum out of your sink, too.
Scrub brushes are also great for getting to floor areas that the mop can't reach, like behind the toilet and sink and between vanities. Brushes with synthetic bristles last longer, but natural bristles are less likely to scratch surfaces.