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: 
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.
The application of the Klein-Gordon equation for the hydrogen atom leads to logical contradictions.
Allowance for relativism in the Klein-Gordon equation should lead to more accurate solutions for the hydrogen atom compared with the Schrodinger equation. However, on the contrary, solutions became increasingly worse with an increase in the nuclear charge. Finally, for the element Z = 69 the solution of the Klein-Gordon equation terminates.
The Klein-Gordon equation has other drawbacks. Therefore, we can say that the Klein-Gordon equation is erroneous.