My sister lives in Whitby so I know it fairly well. Both Whitby and RHB have their own charms and good pubs. If you're only doing one night I'd suggest RHB. There are a couple of decent places for meals in the old village at the bottom of the hill, as well as half a dozen pubs. You could work up an appetite by strolling around the quaint streets or having a walk along the beach. You don't say when you plan to visit. In winter, a lot of places close. Most places are open until after the Oct half term holidays. A nice B&B in the centre of the old village is Orchard House, owned by Mr & Mrs Parry. (The number is in the book).
If you're not familiar with the village, try reading one of Leo Walmsley's autobiographical books (eg The Phantom Lobster).