The December average in Stockholm is 0 ± 2 °C, in Copenhagen 2 ± 2 °C. So the variation from year to year is as big as the difference of the averages. The big difference is rather that it will be noticeably darker in Stockholm.

That's usually in January and February, and far from every year. The climate definitely plays a part, but there is also a cultural difference: Swedes are brought up to respect the dangers of the ice, but that it is your personal responsibility to decide whether it is thick enough (depending on your skill level and equipment) to be safe. Danes – just like Germans – wait for the authorities to give the all clear, which they only do if the thickness approaches 20 cm or so. Kind of funny since it is part of the Danish national mythology that Swedes always depend on their Government to tell them what to do...