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If your main objective is to see large quantities of animals, then the easiest time of year to do this is between June and October - Tanzania's dry season. This is particularly the case in Ruaha and Selous in Southern Tanzania, Katavi in Western Tanzania as well as Mahale Mountains on Lake Tanganyika, although in June and early July the chimps can still be high up in the mountains.
The Ngorongoro Crater Highlands can still be full of flowers in June and July but be prepared for changeable temperatures, it being often cold with low cloud through these months. Tarangire in Northern Tanzania, which acts as a dry season refuge for wildlife from the entire Maasai Steppe is at its best for game between late July to late October, but is always good for elephant and in the green season (December to May) has tremendous birdlife.
If possible, you'll want to avoid visiting during the long rainy season in April and May. The rains are often heavy and sudden, which can result in activities being cancelled last minute.