We are concentrating on more of the southern portion of Germany. My question is, is it better to rent a car or take trains?
"Southern Germany" is not something you can focus on with 10 days - it's just too large and you can't see it all or even drive it all - so I would encourage you to concentrate on specific places, things, and experiences that are important to you.
You ask a rather big question.
Driving isn't difficult in Germany, generally speaking, with the exception of the Autobahn, where you'd better have your wits about you as well as a solid understanding of the driving culture in your head. But for a short visit, driving comes with a lot of drawbacks, especially in winter. A car is useless for the time you will spend in cities of size, which are built around public transport and pedestrians rather than cars - and in December, it's the cities where you will find the most interesting things to see and do as a rule. A trip based on driving around the somewhat bleak countryside from small town to small town, places where the streets are rolled up at dusk (which comes early in December!) may not be very rewarding.
Driving = rental contracts, possible fender-benders and door-dings, insurance concerns, radar-controlled speed zones, difficult parking-fee automats... there are lots of upset rental customers on forum posts like this one who have shared their unhappy surprises and concerns about shady agency practices. Not so with trains.
Any spontaneous sampling of German beer and wine - or the special adult drinks you can get at the Christmas markets - will dictate your driving schedule. Not so with the train (where you can take your drink with you, if you like.)
And if you do have some specific small towns in mind, please don't swallow the idea that you need a car to get there, because you probably do not. Germany's train system is extensive. Let's say you book a stay in Nuremberg. Can you get to smaller towns by train? Here's a rail map of the immediate area around Nuremberg. Zoom in and you'll find dozens and dozens of smaller places very close by and accessible... and others just a little further out:
What about cost? Any day trip you take locally as a couple will cost you €20-32 per day for nearly unlimited use of local trains: