Nuclear fission is the breakdown of large atomic nuclei into smaller elements. Spontaneously fission is due to the fact that the wave function of a large nuclei is 'fuzzier' than the wave function of a small particle like the alpha particle. Induced fission occurs when a free neutron strikes a nucleus and deforms it. Under classical physics, the nucleus would just reform. However, under quantum physics there is a finite probability that the deformed nucleus will tunnel into two new nuclei and release some neutrons in the process, to produce a chain reaction.
Fusion is the production of heavier elements by the fusing of lighter elements.
The process requires high temperatures in order to produce sufficiently high velocities for the two light elements to overcome each others electrostatic barriers. Even for the high temperatures in the center of a star, fusion requires the quantum tunneling of a neutron or proton to overcome the repulsive electrostatic forces of an atomic nuclei. Although it deals with probabilities and uncertainties, the quantum mechanics has been spectacularly successful in explaining otherwise inaccessible atomic phenomena and in meeting every experimental test. This is the holistic nature of the quantum world, with the behavior of individual particles being shaped into a pattern by something that cannot be explained in terms of the Newtonian reductionist paradigm. Newtonian physics is reductionistic, quantum physics is holistic .