Usual invariant theory is dedicated to studying rings; a good example of a result from classical invariant theory is that the ring of invariant polynomials on any representation of a reductive group is finitely generated.

Geometric invariant theory is about constructing and studying the properties of certain kinds of quotients; a good example would be the moduli space of semi-stable vector bundles on an algebraic variety.

In my mind, the difference is this: Classical invariant theory is a collection of results about the interaction between group actions and commutative algebra. Geometric invariant theory is a technique for constructing interesting spaces.