Accessible statement on the intuition and usefulness of group theory to chemistry
Individual molecules rely on point group symmetry - that is, the symmetry about a point. Crystal structures rely on space group symmetry including the translational symmetry in a crystal.

Many molecules, particularly important ones, are highly symmetric. The different symmetry operations (reflections, rotations, etc.) form a finite group.

There are a number of interesting group theory results, but a simple statement comes down to two things:


Understanding Orbitals, Bonding, Vibrations..: Using point groups, any function of the molecule corresponds to a linear combination of irreducible representations. This means I can quickly understand vibrations of a molecule, spectroscopy (as you mentioned) and orbital diagrams and bonding using just pencil and paper.
Faster Calculations: Using symmetry, you can greatly reduce the amount of time required for quantum calculations. For one, you only need to consider the symmetry-unique atoms and bonds. For another, you can show that certain types of integrals and calculations will always be zero and can be completely ignored. This leads to immense speedups.


For example, chemists label orbitals with symmetry labels based on the point group. Using the direct product we can realize that certain interactions will never occur.

Similarly, vibrations have symmetry labels, and we can determine what types of spectroscopy will interact with those vibrations, or whether they are "silent."