Full moon is better in winter because when a part of the Earth is tilted towards the full moon, it's tilted away from the sun and vice versa. So a summer full moon is like a winter sun, only appearing for a few hours and staying close to the horizon.
But a winter full moon is like a summer sun. It's up for a long time and takes a high path through the sky. In the darkest months, the moon, at least when it's full, gives us long, beautifully lit nights.