My understanding from reading reviews and looking at the menus, is that the menus are similar, but Boma has more of an African influence, and Tusker house has more Indian cuisine flavors (curries, samosas, etc). The atmosphere at Boma looks more elegant and upscale (since it's at AKL).
We had dinner at Boma last year and loved it; we have a breakfast ADR there this year. And we loved Tusker House as a CS last year -- sorry to see that go, but we have a lunch ressie there this year, and are looking forward to trying it