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$\newcommand\Top{\mathit{Top}}\newcommand\Mfd{\mathit{Mfd}}$The category can be drastically different. For example, suppose your morphisms are precisely local homeomorphisms. Call this category $\Top^\text{ét}$. This category is locally a topos, which is something certainly not the case for the category $\Top$ of all continuous maps. Moreover, $\Top^\text{ét}$ lacks a terminal object (which is something which will happen for many variants). In fact, $\Top^\text{ét}$ behaves much more like a category of sheaves on a single space, than a category of spaces; to see this, if we let $\mathfrak{Top}^\text{ét}$ denote the bicategory of (étale) topological stacks and local homeomorphisms, this is equivalent to the bicategory of stacks on some (filtered colimit of) étale topological stack(s). The bicategory $\mathfrak{Top}^\text{ét}$ contains $\Top^{ét}$ as a full subcategory, and (if we restrict to a set of topological spaces) is a $2$-topos, so has all the limits, colimits etc. you can imagine. Using this, it can be shown that $\Top^{ét}$ does have at least binary products, but they behave very differently than in $\Top$. For example, instead of $\Top$, consider the category $\Mfd$ of smooth manifolds. Given an $n$-manifold $N$ and an $m$-manifold $M$, their product $N \times^\text{ét} M$ in $\Mfd^\text{ét}$ is empty if $n \ne m$, and if $n=m$, their product is a highly non-Hausdorff smooth $n$-manifold. This is discussed in arXiv:1212.2282.