Signal propagation f(z,t) in an infinite system uniform in z and constant in t may be analyzed in terms of Fourier components f(k,w) as an integral over terms of the form f(k,w)exp(jwt-jkz), where k is wavenumber and w is angular frequency.

The dynamical equation for f(z,t) may resolve into a relation w(k), the dispersion relation. 

In the simplest case of dw/dk (group velocity) constant, propagation of an AM signal with a dominant carrier frequency has amplitude propagating at speed dw/dk, and constant phase fronts travelling at w/k (phase velocity). Attached notes review this.

When dw/dk is not a constant but varies with k, there is dispersion, ie, different frequencies travel at different speeds and a wavepacket f(z,t) will disperse i.e. broaden.

A different situation occurs when dw/dk=0. In this case, the wavepacket does not travel at dw/dk=0; instead propagation is by diffusion if d2w/dk2 is non-zero.  

All of the foregoing apply in the abstract without regard to whether the system is continuous or a numerical model on a discrete grid.

Numerical dispersion refers to dispersion introduced by the finite grid size. For example, the attached exercise explores how the frequency of a 1D "cavity" mode depends on the mesh. With a finite mesh, the numerical model will err in describing frequency and therefore the dispersion relation w(k). This error is more pronounced for shorter wavelengths. The result is numerical dispersion, an artifact of the finite mesh.

Diffusion: Diffusion is the net movement of mols from a region of high concentration to a region of low concentration. "spread out" (if a substance is “spreading out”, it is moving from an area of high concentration to an area of low concentration).

Dispersion: It describes the properties of a wave traveling within a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency.

They are different

Signal propagation f(z,t) in an infinite system uniform in z and constant in t may be analyzed in terms of Fourier components f(k,w) as an integral over terms of the form f(k,w)exp(jwt-jkz), where k is wavenumber and w is angular frequency.

The dynamical equation for f(z,t) may resolve into a relation w(k), the dispersion relation. 

In the simplest case of dw/dk (group velocity) constant, propagation of an AM signal with a dominant carrier frequency has amplitude propagating at speed dw/dk, and constant phase fronts travelling at w/k (phase velocity). Attached notes review this.

When dw/dk is not a constant but varies with k, there is dispersion, ie, different frequencies travel at different speeds and a wavepacket f(z,t) will disperse i.e. broaden.

A different situation occurs when dw/dk=0. In this case, the wavepacket does not travel at dw/dk=0; instead propagation is by diffusion if d2w/dk2 is non-zero.  

All of the foregoing apply in the abstract without regard to whether the system is continuous or a numerical model on a discrete grid.

Numerical dispersion refers to dispersion introduced by the finite grid size. For example, the attached exercise explores how the frequency of a 1D "cavity" mode depends on the mesh. With a finite mesh, the numerical model will err in describing frequency and therefore the dispersion relation w(k). This error is more pronounced for shorter wavelengths. The result is numerical dispersion, an artifact of the finite mesh.

Numerical diffusion refers to the amplitude damping of the numerical solution as compared to the exact solution. 

Numerical dispersion refers to a mismatch in phase between the numerical and the exact solution.

Both phenomena are properties of the numerical scheme employed, regardless of Finite Volume or Finite Difference. One can analyse these properties by employing spectral analysis and investigating the numerical equivalent wavenumber. 

Refer to 'High Accuracy Computing Methods' by T.K.Sengupta, IIT Kanpur for details. It is a wonderful book which lays stress on the numerical properties exhibited by various schemes and how one can decide the range of spatial and temporal stepping by analysing them.

Discrete the governing equations and the terms in the expansion can explain numerical diffusion and dispersion also reveals the physical interpretations of the terms.   

After one discretizes the governing equations of some problem, various schemes may be adopted, each scheme imposes the range of spatial and temporal stepping, eg.: u=u(t), u'(t)=du/dt = (u(t+dt)-u(t))/dt +reste when dt ->0 , here the reste1 (remaining part) is of order of dt (we note reste1 ~ O(dt)). however, we may write u'(t)=du/dt = (u(t+dt)-u(t-dt))/(2dt) +reste2 when dt ->0 , here the reste2 (remaining part) is of order of dt ² (we note reste ~ O(dt ²)). One remarks here that when dt =0.1, we've dt²=0.01, then the second scheme is more accurate than the first one.

The use of first order schemes induces what is called numerical diffusion, because after discretizing governing equation we'll find coefficients like the kinematic diffusivity nu in m²/sec and thermal diffusivity alpha in m²/sec and spatial-to-temporal coef as dx²/dt which is also in m²/sec . the dx²/dt create the numerical diffusion (unwanted). this will introduce a mismatch (deviation) between the numerical and the exact solution, this is called numerical dispersion.

Here the scheme u'(t)=du/dt = (u(t+dt)-u(t-dt))/(2dt) is used as example, the most used scheme as iknow is du/dt = (3*u(t+dt)-4*u(t)+u(t-dt))/(2dt)+O(dt²)

This is regardless of Finite Volume or Finite Difference or lattice boltzmann methods.

hope this is helpful.

Simply speaking, numerical diffusion arises when one approximates first derrivative with corresponding finite difference of odd order of accuracy, e.g. O(h2n+1). Numerical dispersion emerges for finite differences of even order of accuracy. Say, secord order central difference scheme (fi+1-fi-1)/2h would yield spurious oscilations for one dimensional linear wave equation, while first order upwind or Lax-Friedrichs stensils will eventiually smear solution up to complete dissipation.

There are two types of error occurring due to even and odd order of Error Terms.

See Computational fluid dynamics for Engineers volume one by KLAUS A.HOFFMAN 

I remember to have asked this question to my advisor (Javier Jimenez) once and he referred me to the book of Witham "Linear and non-linear waves". Highly mathematical, but easy of read. I suggest you to take a look! 

Interactive book here: https://archive.org/details/LinearAndNonlinearWaves

I guess you question pertains to physical processes.  In that context diffusion is changes of physical property caused by changes in molecular motions whereas, dispersion is change of physical property caused by bulk particle motion.

Diffusion is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Heat conduction, turbulence, and the generation of boundary layer are the result of diffusion in the flow. Diffusion is an equal exchange of species where the final state would be a uniform mixture. An ordinary example would be pouring cream into coffee until diffusion produces a uniform mixture.

Dispersion is the combined effects of convention and diffusion. We talked about smoke dispersion from the chimney, which is result of convective (the wind blowing it), diffusive (smoke diffusive in the air), the bouncy forces (hot air rises).