Beijing is very cold and windy during this time. But also very dry--bring heavy duty moisturizers and hair conditioners. Snow is not common, and when it happens, it's usually only about an inch or less of very powdery stuff. It is possible you will not see any snow during a brief visit in winter. Ice is exceedingly rare. Pollution is not too bad on most days due to wind blowing it away, and clear blue skies are reasonably frequent. You will need full protection gear when you are outside--long underwear, warm socks and shoes or boots, lined gloves, scarf, hat, ear protection. The insides of hotels and public buildings are heated, often overly so (northern Chinese are surprisingly wimpy when it comes to dealing with cold weather). It's similar to Chicago winter but with much less snow. But not as cold as a Green Bay, WI winter (I used to live there!).

Very few tourists either international or domestic at this time, and hotel rates, etc. usually go down. Remember that daylight is short, and most tourist attractions will open later and close earlier. Consider 08:00-16:00 as your useable daytime.

Definitely try to avoid Chinese New Year and even up to one week ahead of it. Many attractions, small restaurants and shops, etc. will be closed, usually for 3 days but sometimes longer. During this time, many more Beijingers and migrants leave for other provinces to see family, than those coming in, so I don't think Beijing and its tourist attractions are particularly crowded during this time--in fact, just the opposite, at least by normal Chinese standards.