The Sombor index and the elliptic Sombor index of a graph \( G \) are defined, respectively, as: \begin{align*} &SO(G) = \sum_{uv \in E(G)} \sqrt{d_{u}^2 + d_{v}^2}, \\[2mm] &ESO(G) = \sum_{uv \in E(G)} (d_{u} + d_{v}) \sqrt{d_{u}^2 + d_{v}^2}, \end{align*} where \( E(G) \) denotes the edge set of \( G \) and \( d_{u} \) is the degree of the vertex \( u \). Since their algebraic forms are closely related, it is natural to ask: Do they share the same extremal graphs? If not, what are the structural differences? In order to address these questions, we investigate \( n \)-vertex trees, chemical trees, and unicyclic graphs, having a fixed number of leaves. For each considered class, the extremal graphs for \( SO \) and \( ESO \) do not always coincide, thereby revealing subtle yet significant structural distinctions between the two indices. We also point out errors in the recent papers [11] and [19], and offer corrections.