Question: Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom?

Why can the first order Dirac equation explain it?
What are the failures?

Answer: 
Electrons are spin-1/2 particles (this qualifies them as fermions, but is not immediately relevant for this discussion).

KG equation is for spin-0 particles, whereas Dirac equation is for spin-1/2 particles. Therefore the proper equation to describe an electron is the latter.

Note that, however, the low-energy limit of both of them gives back Schrödinger equation, therefore in this approximation you can put the spin "by hand" and long as you need it, and this works both for fermions and for bosons.
The reason is electron spin. The energy of the Dirac and Klein-Gordon solution differ precisely by the substitution l<->j.
The Klein-Gordon equation needs to be extended with a term describing the interaction of electron spin with the nuclear potential. The resulting equation is known as the squared Dirac equation. See Itzykson and Zuber, Ch. 2.3, for a solution of the Dirac equation for the single electron case *via* the squared Dirac equation.