Well, you will be highly likely to see most game at the end of the dry season. Still, even if travelling during the wet season, you will see most kind of animals.
June to September is pleasant in Kenya and Tanzania as temperatures are relatively low and there is less rain. January and February are good months as well, albeit a bit warmer. April is probably one of the worst months, with high rainfall.
In the south, September to November is nice, when it has been dry for a while and temperatures are rising but not overly hot. Still, I find prices higher and sometimes accommodations fully booked in these period. June is perfect for me, when temperatures are low, even freezing at night is possible in South Africa and the sun shines almost from sunrise to sunset.
July and August are holiday months in North America and Europe and even southern countries have winter holidays than, like South Africa probaly has a few weeks.
If you combine things, I would go for June or September/October, avoiding rainy seasons, hot weather and crowded places alltogether in both places.