Is it better to stay in the new or old part of town?
Both have their positives and negatives to me. The old town requires more walking to get to most restaurants and commerce (nice big hill to manage).
I would say it also depends on what time of year you are visiting. If in high season, I find it nice to actually get away from the maddening crowds of the main town by heading up to the old town.
Either way, it is a great place to stay.
I've never stayed at Le Beffroi, but I love their garden fior a glass of wine or coffee and their restaurant is very good. One day I might stay there, but we normally just pop over for the day.