What the Klein-Gordon equation essentially distinguishes from Schroedinger's equation is the fact that it is more (or even 100%) like a field equation:
$$(\Box + m^2)\phi = 0$$
Why is this ? First of all, it is a relativistic equation. In a relativistic theory the concept of a wavefunction no longer works since it is not causal.
In Schroedinger's equation the parameter $t$, the time, plays a special role. In a relativistic theory all coordinates of the spacetime should be treated on the same footing. Therefore $\phi$ should be no longer treated as a function of time which describes possible localization of a single particle, but as a function of all four coordinates: $\phi=\phi(t,x,y,z)$.
In case of the interpretation of $\phi$ as a field a probabilistic interpretation of $\phi$ is no longer possible and no longer necessary.
On the other hand a wavefunction makes only sense being complex in order to describe correctly interferences, however, for the solution of a field equation we ask above all for the number of degrees of freedom. For a very simple system we only need one degree of freedom that $\phi$ might describe, so a real solution could be sufficient. For system with 2 degrees of freedom one would choose a complex solution.
Actually, the Klein-Gordon equation can transformed back into a Schroedinger-like equation in the non-relativistic limit. Each $2^{nd}$ order equation can be transformed into a first order equation in time by the definition of new variables. These are for the Klein-Gordon equation (and here the complex numbers come again in):
$$\theta = \frac{1}{2}\left( \phi + \frac{i}{m} \dot{\phi}\right)$$
and
$$\chi = \frac{1}{2}\left( \phi - \frac{i}{m} \dot{\phi}\right)$$
Using $\theta$ and $\chi$ a 2-component matrix equation can be defined like
$$\phi = \left[ \begin{array}{c} \theta \\ \chi \end{array}\right]$$
fulfilling
$$i\frac{\partial \Phi}{\partial t} = H_0 \Phi$$
$\Phi$ is necessarily complex by virtue of its definition.
However, the Hamiltonian $H_0$ still is not a non-relativistic one, since it is non-hermitean which does not allow for a probability interpretation of $|\Phi|^2$. But the form of the equation suggests that the parameter time $t$ seems to have taken over a special role.
But in order to make it really non-relativistic a "Foldy-Wouthuysen" transformation has to be carried out. Only then we get to the non-relativistic limit and all the familiar properties of a solution of a Schroedinger-like equation emerge. More on this specific topic can be read in Bjorken & Drell, Relativistic Quantum Mechanics Vol. 1.
