Both concepts are mathematical in character and they ultimately describe the same characteristics or situations. "Invariance" is a more technical word because it says "what has to be equal to what" for us to say that the symmetry exists.
In particular, the "invariance under a symmetry transformation" means that an object, like the action $S$, has the same value if all the dynamical variables (coordinates, momenta, fields etc.) are transformed according to the symmetry transformation.
The word "invariance" isn't synonymous with "covariance". "Covariance" means that a mathematical object does change but it changes in agreement with the standard rules how a symmetry changes them.
So the Maxwell action is "invariant" under the Lorentz transformations; but the Maxwell's equations are not invariant. Maxwell's equations are formally equations with 4-vectors, if we write them in the relativistic way, and 4-vectors transform as 4-vectors in relativity, "covariantly".
The laws of physics are symmetric under a symmetry group ("group" is the actual technical term in mathematics describing the "type" of a symmetry) if the laws of physics hold if and only if the transformed laws (laws with all the dynamical variables replaced by the transformed ones) hold.
If the laws may be derived from one scalar quantity such as the action $S$, the symmetry of the laws of physics is equivalent to the invariance of this scalar quantity under the transformations.