If you have some specific plan and are not moving constantly, consider booking specific trains directly. This will definitely be cheaper than renting a car for several weeks and will probably be cheaper than an Eurail pass as well. You can read more about elsewhere on this site or on seat61.com but since many trains have restrictions for Eurail pass holders, they don't provide as much flexibility as they used to, especially in France.
An important factor is whether you are traveling alone or not. With a group, traveling by car can be cheaper but if you are alone it's likely to be much more expensive. Depending on where you come from, you might find petrol to be quite expensive (less than in the UK or the Netherlands, though) and you have to remember that most motorways are toll roads (crossing France East-West will cost you about 60 € toll, that's excluding petrol and any other cost for the car). If you need to cover long distances (say go from Paris to the South of France), trains are also likely to be faster and more comfortable so the car's main advantage is really flexibility.
Regarding snow, I don't think it should be a big concern, certainly for Normandy or most of the South and West of France. Unfortunately, I think most rental cars will not be fitted with winter tires so if you are unlucky and there are a couple of snowy days in December, the best solution would be to “sit them out” wherever you are and just delay the next step of your trip for a day or two. Driving with tire chains is only ever necessary in mountainous area when there is a layer of snow on the road, which is almost never the case for long on main roads. If anything, ice or “freezing rain” is more often a problem than snow.
Finally, Normandy as a whole isn't a rural area at all. There are relatively large cities with heavy industry and decent connections to Paris by train. At the same time, if you want to visit the rural or coastal parts of Normandy, a car would certainly be useful and give you a lot more flexibility.