Both get their energy from the difference in mass between the fuel and the end products.
Fission breaks apart very high atomic number atoms, which are naturally unstable. The pieces are medium atomic number, but these pieces have many extra neutrons for their number of protons, and are thus very radioactive. The fission reaction is caused by a neutron colliding with the unstable nucleus, so there is no minimum energy, and fission can and does occur at room temperature, in the proper setup.
Fusion combines two very low atomic number atoms, into more stable versions, such as Hydrogen into Helium. The targets of fusion are often "magic", that is they are somewhat like a ball made of smaller balls that Tesselate nicely in 3D. These fusion targets show on a graph of binding energy per nucleon as a tall spike... He4, O16, and so on. Because the point of fusion is getting atomic nuclei with charge to collide, it requires the atoms move fast enough to overcome the natural repulsion of the Coulomb force long enough for the strong force to interact. This means fusion cannot occur under normal circumstances.
The third possibility for nuclear energy is to make combinations of medium atomic number isotopes with a light isotope, a so-called "low energy nuclear reaction." This is about as believable as alchemy, but so was chemistry at one point. There is some talk about using neutrinos to temporarily cause hydrogen to turn into a neutron, for instance. LENR would be cool if it was real, but right now it just makes for good Science Fiction.
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