The White Man's Burden is a rhetorical mandate to the white man to colonize the other races for the benefit of these same, being their "burden" both the task and the people themselves to colonize. Because of this issue, as well as the resounding title, it soon became an emblem of colonial rule and Eurocentrism.
On the other hand, Rhodes had a mystical concept of imperialism. It owes much to him the conception of the Cape-Cairo axis, which for a long time inspired the British colonial policy and that ended up becoming reality at the cost of the Portuguese, French and German aspirations in Africa.
He wanted British settlers and governors, with local politicians. This put him in conflict with many in Britain, as well as with British missionaries who were not in favor of what was happening in the colony.
In other words, The White Man's Burden proposed a cultural colonization, while Cecil Rhodes only sought the economic exploitation of the territories.