In simpler words symmetry means nature of force and unitarity means Oneness or unchangeability of the force. So symmetry and unitarity are mutually found.

Take this example. A man is driving on a highway from a shopping mall and reaches home. Its the same man that was in the mall that is now at home. So the underlying force didn’t change the man into eg a bear. This “nature” of the objects is called a symmetry and quantitatively this is to be described by forces that are unitary. That is the force thats involved didn’t change the shape and size of the man. shape, size, appearance are therefore unitary. But the man sweat off, changed clothes, lost some pizza weight etc and thus one needs a new force to explain such change. This is a symmetry breaking. The changes are a symmetry breaking. The constancy is a symmetry.

Now think of an electron. The force here is such that the electron will change into something else. It may or may not change, just like the man. In one case a force that describes the electron is a symmetry force. In another its a broken symmetry. Broken symmetries can be added to symmetries to describe the whole picture and this new definition is called unification.

The unitarity is the fact that many variables change but in a way the over all change is zero.

[mainly energy does not change, but also carries to other conserved variable, in Quantum Mechanics probability is connected to energy density hence energy conservation or unitarity is probability conservation, that is, if you normalized probability distribution you will have that so for all time evolution. Normalized probability is 1 which is why the name unitarity. Unitarity force would mean normalized force when all force must add to unit-force ]

When thats so its called a symmetry. electron scattering off a proton is a symmetry or a unitarity group because its still the electron. But electron changing into say a photon by making out with a positron would be a breaking of that symmetry if these two processes are to be defined by two different forces.

Unitarity and Symmetry are the same thing

Broken Symmetry

Spontaneous Symmetry breaking, unitarity, unification