Point-set topology is used to formalize the intuition of continuity and of convergence. It finds its ideal applications for example in Analysis.
The notion of Grothendieck topology is designed to formalize the intuition of patching things together. More technically, this means dealing with descent theory (including sheaf theory of various flavors, e.g. stacks). It finds its applications mainly in Algebraic Geometry, Algebraic Topology, and Logic.
Let's make an analogy. Classical topology is not a direct generalization of metric spaces, rather it's a generalization of some aspects of metric space theory, namely those that pertain to continuity, adherence, etc. To each metric space you can canonically associate a topological space, and not all topological spaces arise in this way (so it's somehow a generalization); but in doing so you lose information since several distinct metrics give rise to the same topology (so it's not really a generalization, rather an abstraction).
In the same way, "Grothendieck topology" is not a generalization of classical point-set topology per se (for example you cannot recover the notion of convergence of a sequence to a point from the G. topology alone, simply because the notion of point is missing), rather it's a generalization of some aspects of point-set topology, namely those that allow you to do descent theory.
Non-homeomorphic topological spaces may induce equivalent Grothendieck topologies: this happens when the corresponding toposes of sheaves are equivalent (talking about categories equipped with a Grothendieck topology is essentially the same as talking about certain categories that are called (Grothendieck) toposes), so in the passage you lose information. Also, there are toposes that do not come from any topological space.
But the situation is slightly better behaved than the passage from metric spaces to topological spaces. Indeed, in topos theory there is the notion of point (which is of course expressed purely in terms of categorical abstract nonsense), and it turns out that if a topos has "enough points" [edit: and is generated under colimits by subobjects of the terminal object, see comment by Denis Nardin] it does indeed come from a topological space in the way you described. Even better, if this is the case, then there is somehow a canonical choice among all those topological spaces that induce the same topos: you take the unique sober one.
This said, I have the impression that the notion of point in the topos context is still quite useless, let alone the notion of convergence of a sequence/net. But I don't know enough about all this to be sure.