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I only have high school physics knowledge, but here is my understanding:

Fusion: 2 atoms come together to form a new atom. This process releases the energy keeping them apart, and is very energetic. Like the sun!

Fission: Something fast (like an electron) smashes into an atom breaking it apart. Somehow this also releases energy. Less energy than fusion, and it's like a nuclear reactor.

Now my understanding is that the lowest energy state is when everything is tightly stuck together (as per fusion), and it costs energy to break them apart..

So.. why do both fusion and fission release energy?

Your assumption about the lowest energy state when everything is tightly stuck together is incorrect.

It only goes this way until you get iron nuclei - and this is why iron is the heaviest element created by fusion.

Creating nuclei heavier than iron consumes energy rather than releasing it. And this is why these elements are only created in supernova explosions and other highly energetic events where there is abundant energy input.

Fission releases energy, because a heavy nucleus (like Uranium-235) is like a cocked mouse trap: it took energy to squeeze all those protons and neutrons hard enough together to make them barely stick (by the nuclear force) against the natural tendency for all those protons to fly violently apart because of their electrostatic repulsion. When struck by an incoming neutron, it is like a mouse touching the trigger pedal of the trap: BANG goes the nucleus.

In the case of fusion, the mechanism is different: the nuclear force between protons and between neutrons is very powerfully attractive but only kicks in when the particles are so close to each other that they are "touching". That attraction is not quite enough to stick two protons together against their electrostatic repulsion but if you add two neutrons to the recipe, you get enough mutually attractive nuclear force to overcome electrostatics and the particles then violently suck themselves together with a very powerful BANG.

Other fusion reactions in which the (2 protons plus two neutrons) get pressed onto a heavier nucleus (like carbon, nitrogen, oxygen, fluorine, ...) release progressively less energy. By the time you get to iron, further fusion reactions actually consume energy instead of releasing it, because the electrostatic repulsion effect gets bigger and bigger- and you are in the province of fission instead.

Fusion:

In a small nucleus there is a relatively large fraction of
nucleons at the surface, which lowers the total binding energy.
The fusion of 2 very small nuclei to one medium-sized nucleus releases energy,
mainly because in the resulting bigger nucleus
there are fewer nucleons at the surface than before.

Fission:

In a big nucleus there is much Coulomb repulsion due to the many protons.
The fission of a very big nucleus into 2 medium-sized nuclei releases energy,
mainly because the total Coulomb repulsion within the 2 resulting
nuclei is smaller than before.

Therefore, medium-sized nuclei (~ 55 nucleons) have the biggest binding energy per nucleon.

The Bethe-Weizsäcker formula for the binding energy of a nucleus
gives a more quantitative explanation for this.