The Hodge theorem (each de Rham cohomology class on a compact Riemann manifold has a unique harmonic representative) has a wide range of applications in complex algebraic geometry, much deeper than showing the finite-dimensionality of the cohomology. One of my favorite results that depend on the Hodge theorem is the Kodaira embedding theorem, which characterizes those compact complex manifolds that can be embedded holomorphically into projective space. See Griffiths-Harris. That a compact manifold has finite-dimensional cohomology groups can be shown in a more elementary way. I am sure this is somewhere in Bott-Tu's book.