To provide a bit of balance, and I know markjohnson already has made his decision based on feedback received, I will offer the following for future readers of this thread, as I live near Bayeux and Caen and visit both several times a year for shopping or other reasons.
Bayeux was not the scene of a battle in WWII, so its central area remains largely as it was before the war. In effect, it's a specimen of how much of Normandie appeared before the war. And that makes it interesting to visit. There are a multitude of these kinds of towns in France, but not in Normandie; where many of them were destroyed during the war.
Caen was the site of protracted fighting between the Germans and the British in June and July 1944. A substantial fraction of the city was destroyed in that fighting and had to be rebuilt in the decades after the war.
Caen is diverse, much, much larger than Bayeux, and has far more cultural offerings than Bayeux.
Bayeux is small by comparison to Caen but not small by Norman standards, where towns of 8,000 to 10,000 population are considered pretty large.
There are small parts of Caen that retain historical appearances. The Abbaye aux Dames de Caen and the area around it spring to mind. But mostly, I would recommend Caen if you want to experience a modern French city with all the cultural and dining options such as place might offer.
Bayeux has far more limited offerings in that regard. It has a few good restaurants. But it's more a quaint showpiece that attracts tourists than it is representative of a real French city.
It all comes down to what you want to experience.
