| A fission weapon is, baldly, an atomic bomb in its original form. The term describes the device's mode of operation and purpose; it is a pure fission device, and has been weaponized, and contains no design or materials intended to 'enhance' the fission output, relying purely on the energy released from the initial mixing of fissiles to critical mass. Fat Man and Little Boy were both fission weapons; since their construction, other variants of 'The Bomb' have been produced for various purposes.
The advantage of the pure fission weapon is its simplicity; the design can be overengineered to the point where testing isn't even necessary. Little Boy was a gun-type fission weapon, firing a slug of uranium-235 into a larger block of the same material in order to reach critical mass, and trapping an initiator (made, I believe, of polonium) inside the mass to provide the neutrons to start the ball rolling.
Fat Man, on the other hand, was an implosion device, which was more complicated and required testing. It was pursued because despite the device being bigger (initially) the actual physics package was much more efficient, and given the incredible expense of producing fissionables, that alone made it more desirable. The size of the device was mostly given over to components which could quickly be miniaturized once it was proven to work (and were, rapidly) whereas the Little Boy design was less amenable to shrinkage. Furthermore, the implosion system, once minaturized, was more easily adapted as a 'primary' for multistage weapons.
The disadvantages of the pure fission weapon are its size and its requirements. Since pure uranium fission isn't nearly as efficient at producihng neutrons as other options, you will need a large amount of fissile material to get a supercritical reaction. As a consequence, the weapon itself will be bulky and extremely heavy. Both of these are Bad Things(tm) to the aspiring weaponeer; usually because fissile materials are an extremely scant resource, and because the bigger the weapon the more difficult it is to deliver.
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