Which is cheaper? Nice or Barcelona?

I expect that each city will have a similar range of prices, with your actual cost tied to the one boat and one house that you reserve. You might want to shop around for those before making the flight arrangements or other decisions that will follow on from your selection. An un-touristy town would be less expensive but perhaps hard to find along the coast.

Will this be for a summer stay?

I think you may find more month-to-month price variation in Nice than in Barcelona, though I have never researched real estate rental costs in either city. The peak period for hotels in Nice is July and August, with even June being substantially less expensive.

Sightseeing in Nice and the surrounding towns is generally less expensive than in Barcelona, where many of the most popular sights cost over 20 euros per person. Around Nice, though, you would be incurring transportation costs.

when is the trip?

The bigger the house you seek, the more likely you are to be in a villa and chalet outside whatever town you choose. Those accommodations often rent by a minimum of a week, although no doubt there are exceptions.