These are indeed related. The first thing to know is that they both ``live'' (i.e., sheafify) over the critical locus (this is not saying much if you assume $W$ has isolated critical points, but interesting in more general cases). This is clear for $\operatorname{Coh}(\operatorname{Crit}(W))$. For matrix factorizations, we can identify $\operatorname{MF}(W)$ with the category of singularities $\operatorname{Coh}(W^{-1}(0))/\operatorname{Perf}(W^{-1}(0))$, which vanishes on the regular part of $W^{-1}(0)$.

In the case where the superpotential $W$ is Morse, or Morse-Bott, the category of matrix factorizations is close to being the $2$-periodization of $\operatorname{Coh}(\operatorname{Crit}(W))$. This is the subject of Teleman's Matrix Factorisation of Morse-Bott functions. 

The basic case to understand is when the superpotential is $W = x^2$ defined on $\mathbb{A}^1$. The category of matrix factorizations is $\operatorname{Coh}(k[x]/x^2)/\operatorname{Perf}(k[x]/x^2)$. A Koszul duality computation tells us that this is the category of modules over the algebra $k[\beta, \beta^{-1}]\langle \eta | \eta^2 = \beta \rangle$. Here $\beta$ is a variable of cohomological degree $2$ giving the $\mathbb{Z}/2$-grading, and $\eta$ is a variable of degree $1$. We interpret this result as a $2$-periodic Clifford algebra on a generator $\eta$ of degree $1$. The derived critical locus in this case is just $\operatorname{Spec}(k)$. You can then think about $\operatorname{MF}(x^2)$ as the $2$-periodization of $\operatorname{Coh}(\operatorname{Spec}(k))$, with an extra Clifford factor.

From this you can build higher dimensional Morse functions: the rule is that $\operatorname{MF}(f \boxplus g) = \operatorname{MF}(f) \otimes_{k[\beta, \beta^{-1}]} \operatorname{MF}(g)$ (see Preygel's Thom Sebastiani and Duality for Matrix Factorizations). The tensor product of $k[\beta, \beta^{-1}]\langle \eta | \eta^2 = \beta \rangle$ with itself turns out to be $k[\beta, \beta^{-1}]$. With this one can conclude that when $W = x_1^1 + \ldots x_n^2$ the category of matrix factorizations is the $2$-periodization of $\operatorname{Coh}(\operatorname{Crit}(W))$, with an added Clifford factor in odd dimensions.  In the Morse-Bott case one has extra corrections arising from the topology of the normal bundle to the critical locus.

When you move beyond the Morse-Bott case the two categories start diverging. Their relationship in general is roughly like the relationship between the algebra $D_X$ of differential operators on a smooth scheme $X$, thought of as the deformation quantization of $T^*X$, and $\mathcal{O}_X$. The derived critical locus of $W$ has a $(-1)$-shifted symplectic structure. If one works in the $2$-periodic context this turns into a $1$-shifted symplectic structure, which allows one to deform the symmetric monoidal category $\operatorname{QCoh}(\operatorname{Crit}(W))$ into a monoidal category. This is the monoidal category of modules over the $E_2$-algebra associated to the Gerstenhaber algebra $\mathcal{O}_{\operatorname{Crit}(W)} \otimes k[\beta, \beta^{-1}]$. This $E_2$-algebra turns out to agree with the Hoschild cochains of $\operatorname{MF}(W)$, which implies that the deformation of $\operatorname{QCoh}(\operatorname{Crit}(W))\otimes k[\beta,\beta^{-1}]$ acts on (the ind-completion of) $\operatorname{MF}(W)$.