I don't think that Grothendieck topologies should be viewed as analogous to ordinary topologies. It is true that a topology on a set induces various Grothendieck topologies on various categories, but so does every system of basic open neighborhoods. In my opinion, it is more fruitful to think of a Grothendieck topology as the analogue of a system of basic open neighborhoods and a topos as the analogue of a topological space.
Let me try to answer the following question, which may or may not be the question that is actually being asked: Why do we prefer topological spaces to systems of basic open neighborhoods, and topoi to Grothendieck topologies? I think that the answer has to do with morphisms.
To give a morphism of spaces with basic open neighborhoods, one must give a function that respects those neighborhoods. One can't, however, require that the pre-image of a basic open neighborhood be open. Instead, one has to require that each point contained in the pre-image of a basic open neighborhood have a basic open neighborhood inside the pre-image.
Not only is this definition more complicated than the one for topological spaces (and the extension to Grothendieck topologies is by no means obvious!), but there are multiple distinct but isomorphic systems of open neighborhoods on the same space. A topology is a maximal system of open neighborhoods in a given isomorphism class, which makes it a "best model" for a particular notion of nearness.
Another interpretation that makes this model appealing is that given a system of basic open neighborhoods, the open sets are the "local properties" (those that hold at a point if and only if they hold at all sufficiently nearby points). (If one believes a slogan like "there are two -1-categories: TRUE and FALSE" then open sets are "-1-sheaves"; this completes the formal analogy with Grothendieck topologies and their associated topoi, which are their associated categories of "0-sheaves".)