There's nothing to see in Weymouth from the days of the Black Death - but it's a pleasant enough tourist town, and a good way to access Dorset's Jurassic Coast which is nearby - a must for fossil hunters, and a World Heritage Site.
Norwich, on the other hand, oozes medieval history. It's chock full of churches from the Middle Ages. The Castle, which overlooks the town, is more impressive from outside than in. But the Cathedral is wonderful; the cloister is one of the most lovely in England.