All seasons have their own beauty and specialty. A few practical issues to take into account:
Game viewing in the Serengeti is always a highlight, but the animals travel according to season. December-April: eastern and southern Serengeti (March-April: new wildebeest babies are born). May-July: western Serengeti. June-July: the best time to see the wildebeest migration in the Serengeti. This is a very special event, but also leads to more tourists and higher prices. September-November: Northern Serengeti.
April-May are officially low season, with low season prices of the lodges and less tourists but still perfect game viewing opportunities.
The rainy season starts in November- December with the long rains and continues in March-April with the short rains. The roads in the northern circuit are very good, but if you are planning to visit other parts of Tanzania, rainy season might not be the best season to visit. It will rain daily in rainy season (but never the whole day), but the advantage is that the land will be lovely green and the bird watching opportunities are many. You are able to see the resident species as well as the migrant birds that live elsewhere in the dry season. The temperature in rainy season is slightly higher, which is hardly a problem during your safari, but can be a problem if you are planning to visit the humid coast of Tanzania.
The dry season is important if you are planning to climb mount Kilimanjaro or mount Meru. The best period would be December to February or August to September. Also temperatures on the coast and in Zanzibar tend to be better. Dry season makes the parks Manyara National Park and Tarangire National Park an oasis for animals. Big herds of elephant, zebra and buffalo will be attracted by the last sources of water in the lake and the river.