Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition.
In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law.
In crystals, the electron density is periodic and invariant with respect to discrete translations by unit cell vectors.
For example the rule describing Newton's force of gravity between two chunks of matter is the same whether they are in this galaxy or another (translational invariance in space).
The law does not work differently depending on whether one chunk is east or north of the other one (rotational invariance).