For the noun referring to a flat grassland in a tropical or subtropical region, savanna is the usual spelling in American English, while savannah is preferred in all the other main varieties of English.
Although the h is usually left off in U.S. writing, the city in the U.S. state of Georgia is spelled Savannah, as is the river on which the city lies.
There’s no need to capitalize savannah in the phrase African savannah or in similar phrases, as there is no single African savanna but rather several distinct regions to which the word can apply.
Examples
U.S.
Twenty-two dead elephants, including several very young ones, clumped together on the open savanna, many killed by a single bullet to the top of the head. [New York Times]
[I]t may have enabled our ancestors to keep cool, for example, while traveling across the hot savanna. [Slate]
In the pine savannas of New Mexico’s high country, the natural regime was one of frequent, low-burning wildfires … [Los Angeles Times]
Outside the U.S.
The Pokomo had earlier killed 52 members of the Orma tribe in the Tana River Delta, a region that includes savannah. [CBC]
He had been a hardened enforcer for the Queen Empress Victoria, traversing desert, jungle and savannah. [Daily Mail]
His favourite destination is the northern rangelands, a rolling savannah that spreads north from the snow-tipped silhouette of Mount Kenya . [Sydney Morning Herald]