I'm enrolled on a short and conceptual couse on RQM and QFT and the professor made a distinction about the Klein-Gordon (K-G) equation on RQM and the K-G equation on QFT. Roughly speaking, he said that RQM is about particles and QFT in about fields, and the object which captures that distinction is in fact the wave function. Then he said that the wave function on K-G of RQM is the common wave function from non-relativistic quantum mechanics, and the "wave function" from QFT is, also, a scalar field. He said that in RQM the wave function $\psi$ is for one electron and in QFT in for the field $\Psi$ of electrons, but since the wave function $\psi$ is already an scalar field, in the sense of pure mathematics, how can describe one electron and not a field?
Relativistic quantum mechanics (RQM) constructs single-particle wave equations that are consistent with special relativity (SR). However SR allows to create particles out of energy, while quantum mechanics is based on the conservation of probability. Nevertheless at energies low compared to the masses involved a single-particle description is a good approximation to nature.
Quantum field theory (QFT) enables the creation and annihilation of particles. The classical fields are promoted to operators that create and destroy particles. These operators commute with each other if they have integer spin, or anticommute for half-integer spin. A scalar field has spin zero, it commutes. Relativistic covariance is built in. This approach is known as second quantization.
Therefore there is a conceptual difference between the wave function in RQM and the wave function in QFT.