Definition of Thermal Diffusivity and Thermal Conductivity
Definition
$$ {\frac{ \partial }{ \partial t }} \mathbf{u} = \alpha \nabla^2 \mathbf{u} $$ In the above general heat equation, the heat diffusion coefficient $\alpha$ is called the thermal diffusivity. The product of the thermal diffusivity $\alpha$, the density $\rho$, and the specific heat $C_{p}$ is called the thermal conductivity. $$ k := \rho C_{p} \alpha \left[ {\frac{ \mathrm{W} }{ \mathrm{m} \cdot \mathrm{K} }} \right] $$
Explanation
According to dimensional analysis, the dimension of the thermal diffusivity $\alpha$ is $\mathsf{L}^2 \mathsf{T}^{-1}$ and its SI unit is $\mathrm{m}^2/\mathrm{s}$. Indeed, in the heat equation the left-hand side is differentiated once in time while the right-hand side is differentiated twice in space.
$$ \rho C_{p} {\frac{ \partial }{ \partial t }} \mathbf{u} = k \nabla^2 \mathbf{u} $$ If the heat equation is expressed in terms of thermal conductivity, it appears as above. Physically that form is more intuitive, but algebraically it is usually more convenient to work with the thermal diffusivity.
Comparison between thermal conductivity and heat transfer coefficient
The SI unit of thermal conductivity $k$ is as follows. $$ k \left[ {\frac{ \mathrm{W} }{ \mathrm{m} \cdot \mathrm{K} }} \right] $$ On the other hand, the SI unit of the heat transfer coefficient $h$ is as follows. $$ h \left[ {\frac{ \mathrm{W} }{ \mathrm{m}^2 \cdot \mathrm{K} }} \right] $$ Unlike thermal conductivity, the heat transfer coefficient is expressed per unit area. The heat transfer coefficient quantifies heat transfer between an object’s surface and the environment; since a larger surface area allows heat transfer to proceed more readily, expressing it per unit area is more intuitive.