Let $\mathcal{X}$ be a non-empty set. For instance, let it be a set of vectors of the form $\mathbf{x}\in\mathbb{R}^n$, i.e., $\mathcal{X}=\{\mathbf{x}_i\mid\mathbf{x}_i\in\mathbb{R}^n,\:i=1,\ldots,m,\:m\in\mathbb{N}\}$.
What is the difference in requiring that $\mathcal{X}$ is a) a subset ($\subseteq$), and b) a proper (or strict) subset ($\subset$) of $\mathbb{R}^n$? That is, what is the difference between the following expressions $$ \mathcal{X}\subseteq\mathbb{R}^n, $$ and $$ \mathcal{X}\subset\mathbb{R}^n. $$ Apologies in case the question is too vague, or the answer is too obvious!
The only distinction to be drawn is whether $\mathcal{X}$ is allowed to equal $\mathbb{R}^n$. If we really want to make it clear to the Reader this is not allowed (a proper subset, as you said), the notation $\mathcal{X} \subsetneq \mathbb{R}^n$ might be better.
In the case described in the top paragraph, the condition seems to be imposed that $\mathcal{X}$ is countable, and in that circumstance it would be impossible for equality to exist (between a countable and an uncountable set). Then the notational distinction would be moot.