In addition to the answers above, I would like to add what goes wrong if you try to use Klein Gordon equation for higher spin. In this case, you always get ghost particles, you get particles whose contribution to the probability is negative.
This issue is best explained by trying this trick with a vector. Consider a four-component vector, with each component separately obeying the KG equation. In this case, you get
$$ S = \int \partial_\mu A_\nu \partial_\mu A_\nu - {m^2\over 2} A_\mu A\mu d^4x$$
and the kinetic term on the time component of A is opposite sign to all the space components. This means you have a timelike ghost field, or three spacelike ghost fields, and the equation is not describing a reasonable free field theory.
Nevertheless, this thing (in the massless case) is what you get in QED when you impose Feynman gauge (or in t'Hooft gauge for the nonabelian theory). The ghost time component doesn't contribute to physical processes by gauge invariance, and this means that the negative propagator states are not propagating degrees of freedom.
This disease affects the KG equation with any spin other than 0, so the other wave equations are the proper ghost-free trucations of the Klein Gordon equation, which is best viewed as only part of the equation of motion--- the mass shell projection.