Wikipedia (and many other sources) say that by extending the number of spacetime dimensions from four to five, Kaluza–Klein theory unifies general relativity and electromagnetism into a single theory. But why do need to do anything as complicated as adding a new dimension? It's conceptually very easy to write down Maxwell's equations in curved spacetime without doing anything nearly as fancy. How is this any less "unified?"
In the quantum context, a compact extra dimension gives a natural justification for the quantization of electric charge, but Kaluza's original theory was entirely classical.