In Material Science, what is a primitive cell when speaking crystals.
Primitive unit cell contains only one lattice point while conventional unit cell have more than one point, and as said by M Quack are usually considered for symmetry purpose.
You can choose a Wigner-Seitz cell that is primitive and has the full symmetry of the lattice. I think one reason of using conventional unit cells is that it sometimes makes simpler the calculations.That is how is defined in my book. Thus, the disadvantage of using a primitve cell is that how you can only have one lattice point inside of, sometimes you will get a pretty weird shaped cell. If you are more flexible, and if you decide to have more lattice points inside of your unit cell, you can get a better shape. For instance, if you use a cube with for lattice points, you can describe the Bravais lattice of a FCC very nicely. However, if you try to describe this very same lattice with a primitive cell, you will get a very weird thing.
Please, let me know if that helped you. I am also studying this subject, so we could discuss it together in more depth.