Briefly, the difference in the bombs dropped on Hiroshima and Nagasaki reflect the fact that the US pursued two different types of nuclear fuel (fissile material) production during the war, not knowing which one would be successful (if either). At their site in Oak Ridge, Tennessee, the Manhattan Project erected massive factories for the enrichment of uranium, while at their site in Hanford, Washington, they build nuclear reactors and processing facilities that allowed the breeding of plutonium. Between the two of these facilities, they ended up, by coincidence in effect, with about 64 kg of enriched uranium and about 13 kg of plutonium by mid-July 1945.

They had initially planned to use a simple, no-possibility-of-failure weapon design for the weapons known as a gun-type design, where you just shoot one piece of fissile material into another, which with the right sized pieces and in the right kind of configuration can produce a nuclear explosion. They discovered in the summer of 1944, however, that this approach cannot be used with reactor-bred plutonium (it has a high background rate of neutron activity, which would cause it to prematurely detonate with a much small explosion if you tried it). The scientists reorganized their research design and produced a second bomb design, the implosion design (in which high explosives are used to compress a solid ball of fissile material to several times its original density), to work with plutonium.

The implosion design was more complex and had a higher chance of partial or complete failure, so they tested one version of it in mid-July 1945 (the Trinity test). It was more successful than they had guessed it would be.

So at the end of July 1945 they had enough uranium for one gun-type bomb and enough plutonium for one implosion-type bomb. They did briefly consider taking the uranium from the gun-type bomb and converting it into an implosion-type bomb, which would be far more efficient in terms of fuel usage (e.g. it would allow them to have many more bombs in their arsenal), but would take several weeks to enact. The military decided to go with the weapons they had in hand. The uranium bomb was dropped on Hiroshima, the plutonium bomb was dropped on Nagasaki.

So to recap: the differences in the weapons came down to differences in fuel, which dictated different designs (at least in the case of plutonium), combined with choices made about expediency. The differences were not part of some bigger war strategy, in other words, but a result of the uncertainties involved in making the first nuclear weapons (since they didn't know which method would actually work, if either, when they started).

A good general overview of all of this for a popular audience is Richard Rhodes' The Making of the Atomic Bomb.