The weather in December is usually good; it's hot but not too hot to enjoy walking, safari etc. November and December in Tanzania can be rainy; it's the time known as the short rains, so less heavy than the March rains. While June to October is usually considered the ideal time to visit, December can be a very good time too as a lot of the wildlife, eg wildebeest has returned to the Serengeti plains by this time. Especially in the north of the country, there is a great deal to interest birdwatchers. Also from December onwards, in the south there are great hiking opportunities especially around Loliondo. Equally, the hot weather makes it very pleasant to opt for a seaside trip - safari and beach holidays combined are very popular.
You may like to read the experiences of some other travellers on safari in Tanzania - you can find a discussion here.