Schrödinger equation derivation and Diffusion equation

The Schrödinger equation is a wave equation, not a diffusion equation. While the equations look similar, the $i$ in Schrodinger equation differentiates them; that allows non-decaying oscillatory solutions, which diffusion equations do not allow.

That said there are certainly relations between the two.

The Schrödinger equation is analogous to the Fokker-Planck equation, which is the evolution of a classical probability distribution subject to random noise. That can result in diffusion.

There is also the stochastic interpretation of quantum mechanics, which relates the Schrödinger equation to a kind of quantum Brownian motion. (Truthfully, I don't understand it; the original paper is here.) Classical Brownian motion leads to diffusion.

I don't know whether Schrödinger proved or guessed the equation with his name, but this equation can be derived similarly with the diffusion equation - see Gordon Baym, "Quantum Mechanics".

However, differently from the diffusion equation, the diffusion coefficient in the Schrodinger equation is imaginary. That tells us that we have to separate the Schrödinger equation into two, one equating the real parts of the two sides, and one equating the imaginary parts. The meaning of this imaginary diffusion coefficient is therefore that the wave-function is complex, or, in other words, it has an absolute value and a phase, like the electromagnetic wave.