What is the difference between the strong and weak nuclear forces of the universe?

The strong nuclear force is responsible for binding protons and neutrons together in an atomic nucleus. It is strong and short ranged and has to overcome the electromagnetic force which is pushing positively charged protons apart.

A good example of the strong force is the fusion process which happens in smaller stars such as our sun. Positively charge protons repel each other. At the extreme temperatures and pressures in the sun's core, two protons can get close enough together for the strong nuclear force to bind them into a bi-proton or Helium-2 nucleus.

A bi-proton is very unstable and most of them fly apart. For the fusion process to continue to produce Deuterium the weak nuclear force is required.

The weak nuclear force is responsible for radioactive decay by being able to convert a proton into a neutron of vice versa. To be more precise it converts an up quark to a down quark or vice versa by means of the W boson. In the case of fusion a proton is converted into a neutron, a positron and an electron neutrino.

In fact the strong nuclear force doesn't really exist. Early theories described the strong nuclear force as binding protons and neutrons using the pion as the force transmitting boson. We now now that protons, neutrons and pions are composite particles consisting of quarks bound by the colour force transmitted by gluons. So, the strong force is actually a residual effect of the colour force extending beyond the inside of protons and neutrons to bind them together.

The strong nuclear force is responsible for holding protons and neutrons together within the nucleus of an atom, while the weak nuclear force is responsible for certain types of radioactive decay, such as beta decay.