Let $\nabla^λ$ denote the Schur functor labelled by the partition $λ$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^λ\!\mathrm{Sym}^\ell \!E \cong \nabla^μ\!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $λ$ and $μ$ are conjugate partitions and when one of $λ$ or $μ$ is a rectangle. We give a complete classification when $λ$ and $μ$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^λ\!\mathrm{Sym}^\ell \!E$ is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon's enumeration of plane partitions, and prove a new $q$-binomial identity in this setting.