The earliest that I've been in Antibes/Côte d'Azur has been March. It was sunny and 60s/low 70s Fahrenheit. Antibes has a slower pace, fewer people than Nice. Nice is bigger, more vibrant.
While you don't need a car for staying in Antibes, it's great for visiting the surrounding hill towns, such as Mougins, Tourettes-sur-loup, Éze Village, St Paul-de-Vence, etc.
Along the coast, you can take the train and visit different villages...Cannes, Antibes, Biot, Nice, Villefrance-sur-mer, Monaco, Menton, etc.
As Pauline mentions, you can walk around the Cap from Old Town, and you can also walk out to Fort Carré, which has a very long gravel path around it.
I'm not sure of the restaurant openings/closings in January for the coastal cafés (such as on La Garoupe beach on the Cap).
You'll need a car if you want to drive the coastal highway from Cannes to St Tropez. There's one road into St Tropez from the east, and on weekends, it can be crowded. I've driven to St Tropez twice to hike the peninsula, and turned around once when the traffic was jammed. Along the way, you can stop in Frejus and/or St Raphael.
If you want to drive the Grand Corniche above Nice, you'll also need a car. Great views. Not sure of the road conditions in January.