The definition I have says
An ensemble index by $I$ is a sequences of random variables indexed by $I$. Namely, any $X = {X_i}_{i \in I}$, where each $X_i$ is a random variable, is an ensemble indexed by $I$.
How is this different from just a set of random variables?
Simply put, a set isn't indexed, it's just a collection of unique elements.
Without context, I don't know why it serves the author's purposes to define this structure as opposed to just considering a set of random variables, but presumably it is helpful for an argument they are making.