Why isn't the set of all $m\times n$ matrices over $F$ denoted ${F^m}^n$? Is there some technical reason that this notation is not used, or is it just a convention to use $\mathrm{M}_{m\times n}(F)$ instead?

It’s not hard to prove $F^{mn} \cong M_{m\times n}(F)$ as vector spaces, or rings (in the case $m=n$); it's only a matter of notation. However, notation is a matter of convenience and things should be written clearly.
The first one is not very clarifying: you could add a matrix ring structure to $\mathbb R^4$, but reading it as $M_{2\times2}(\mathbb R)$ makes it easier to think about the set with operations as matrices, since the notation speaks about the underlying algebraic structures. If you're talking about vectors, you shouldn't use $M_{2\times2}(\mathbb R)$, and if you're talking about matrices, it doesn't make sense to use $\mathbb R^4$.
