Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication.

It is known that the determinant of an $n\times n$ matrix can be computed in $\tilde{O}(M(n))$ time, where $M(n)$ is the minimum time required to multiply any two $n\times n$ matrices.  It is also known that the best circuit complexity of determinants is polynomial at depth $O(\log^{2}(n))$ and exponential at depth 3.  But the circuit complexity of matrix multiplication, for any constant depth, is only polynomial.

Why is there a difference in circuit complexity for determinants and matrix multiplication while it is known that from an algorithm perspective determinant calculation is similar to matrix multiplication? Specifically, why do the circuit complexities have an exponential gap at depth-$3$? 

Probably, the explanation is simple but I do not see it. Is there an explanation with 'rigor'?

Also look in: Smallest known formula for the determinant
Consider circuit value problem and Boolean formula evaluation for various small complexity classes. Deterministic sequential time complexity of them are the similar as far as we know, yet they are very different from circuit complexity perspective. Similarity in one particular type of resource on one model doesn't imply similarity for other resources in other models. One problem can be such that we can exploit parallel computation for one while we can't do that for another one, yet their sequential time complexity can be the same.

When can we expect a more robust relation between the complexity of two problems across models and different resources? When they're are robust reduction between them in both directions that respects resources in those models.

Edit: multiplication has subexponential size depth 3 circuits. Proving a lower-bound of that kind for determinant would show it is not in $\mathsf{NL}$ separating it from $\mathsf{NC^2}$ which is not known.