Why were the atomic bombs Fat Man and Little Boy designed so differently?

When the Manhattan Project began, they knew of two ways of making nuclear fuel — enriching uranium and breeding plutonium — but didn't know which of those two approaches would be successful within the constraint of the project (which was mostly time). So they invested in both, in parallel, building massive facilities at Oak Ridge, Tennessee, to enrich uranium, and the first industrial-sized nuclear reactors at Hanford, Washington, to breed plutonium.

They initially were very conservative in the design of the bomb, preferring a simple "gun-type" method of explosively combining two subcritical masses of fuel into a single critical mass. They had other ideas on the table from the beginning, but they all seemed much harder to calculate and engineer than the gun-type approach, and again, time was their main consideration.

By the summer of 1944, both the uranium enrichment and plutonium breeding were getting slowly underway, and they obtained the first samples of reactor-bred plutonium. They discovered, to their horror, that it would not work in a gun-type bomb. Because reactor-bred plutonium contains an undesirable contaminant (the isotope Pu-240, along with the desired isotope Pu-239), its background neutron rate was far too high. A gun-type plutonium bomb would pre-detonate before creating a significant explosion.

Their options at this point were to just have enriched uranium bombs, but it took so long to make enough uranium for a weapon, and took so much fuel for a gun-type design, that they would only have one available every two months once their production line got up and running successfully. Instead, they threw themselves into another design of a bomb for the plutonium, an "implosion" design, which involved using carefully-sculpted high-explosives to compress a solid ball of plutonium to several times its original density. This would also create a critical mass. This was much, much more difficult to do than the gun-type bomb, and the entire Los Alamos laboratory was reorganized around solving this difficult problem.

It was so difficult that they were not satisfied that it would necessarily work, even after they finished it. With the gun-type design, they could do enough non-explosive testing to have confidence that it would work and would be of sufficient explosive power for their needs. But the implosion design relied on many very-hard-to-calculate factors, and so they felt they needed a full-scale test in the summer of 1945 (the Trinity test, July 16, 1945).

After Trinity, they knew the implosion design worked very well. There was a suggestion by Oppenheimer that they might disassemble the uranium for the gun-type bomb (Little Boy) and instead use it to manufacture "composite" implosion bombs (Fat Man) that contained both enriched uranium and plutonium, which would have increased their ready-to-use arsenal by quite a lot. But because that would involve some minor delays to the first use of the bomb, this was vetoed and the original plan — to use the uranium bomb first (Little Boy) and the implosion bomb second (Fat Man) was put forward.

So, in brief, they are the result of two different production directions, and an uncertain scramble to be able to use plutonium in an atomic bomb. The gun-type bomb was obsolete after the Trinity test but was used anyway because they didn't want to risk further delays.