What is the difference between the diffusion equation and the heat equation? I know that the diffusion equation is a more general version of the heat equation. But what is the exact difference informally?
 A: The difference is typically the diffusion coefficient:
\begin{align}
\frac{\partial \psi}{\partial t}&=\nabla\cdot\left(\kappa\nabla\psi\right)\tag{diffusion}\\
\frac{\partial \psi}{\partial t}&=\kappa\nabla^2\psi\tag{heat}
\end{align}
Under the diffusion equation, we typically take $\kappa$ to be a spatially-dependent variable whereas in the heat equation it is a uniform constant (allowing us to use the Laplacian on $\psi$). 
However, the heat equation can have a spatially-dependent diffusion coefficient (consider the transfer of heat between two bars of different material adjacent to each other), in which case you need to solve the general diffusion equation.
There is no relation between the two equations and dimensionality. Both equations can be solved in one dimension, with a straight-forward substitution $\nabla\to\frac{\partial}{\partial x}$, or left in the multiple dimensions as I give above (which would likely require a numerical solver).
A: There is no difference physical or mathematical .Heat equation is ONE application of the diffusion equation whether one,two or three dimensional and whether the diffusion coefficient is spatially uniform or not.No difference also between both in considering or accommodating the source/sink term. 
A: Both diffusion and heat conduction lead to the same mathematical problem. The only difference are the physical dimensions of the quantities in the equation and their interpretation (temperature vs. concentration/probability).
The analogy can be seen beautifully in irreversible thermodynamics where you obtain the equations by combining a conservation equation for either energy or concentration with a linear relation between thermodynamic fluxes and forces, Fourier's law or Fick's law, respectively.
Mathematical terminology may vary between authors. For example, Evans (PDE) calls one and the same equation both "heat equation" and "diffusion equation".
