Neuromancer is a particular version of the quest genre—what we'll call the heist genre. See, a quest is all about a hero traveling to reach a certain goal, usually involving him finding an object of immense power or value. Ever seen an Indiana Jones film? Every one of those movies is a quest story where Indiana Jones travels to a foreign land to find either the Ark of the Covenant or the Holy Grail or some weird plastic alien head.
The heist genre's a lot like that, only instead of a hero you have an antihero—a criminal or thief or hacker—trying to acquire a valuable or powerful object, probably by some shady means. In Neuromancer, our antihero is Case, a hacker trying to steal Neuromancer (the object). Just like Indiana Jones, Case travels to various locales like Japan and outer space to reach his goal. In this way, Neuromancer shares a lot in common with other heist stories such as Ocean's Eleven and The Sting.
As for science fiction, well, it's science fiction. It takes the science of today and considers what it will be like in the future. As a result, we have the Internet replaced with the virtual reality of cyberspace. We also get our current space exploration advanced to people actually living in space. Plys, people change their bodies with some seriously cool technology (as opposed to mere botox). Oh, and every bookstore ever shelves the novel in their science fiction section—kind of a giveaway that.