I used a trailer to pull my kids to daycare and school on a daily basis for over seven years and I used it a few times in the snow. If it was choice between a child seat or trailer I would choose trailer. With that said, it can be difficult in the snow. Your brakes are the biggest issue. Rim brakes work poorly when they get full of ice and snow. They only way I would pull a trailer in the snow (if I had to do it again) is if I had a disk brake equipped bike.
On my first ride in the snow with the trailer, I slipped on some street car tracks and slid along the road into the path of an oncoming street car. I was lucky that I jumped up fast enough to get my bike and the trailer out of the way. Street cars don't stop well in the best of times. This was enough to scare me and only use the trailer on clear road days.