Genres in fiction are established due to a need/want from an audience. It's only as a particular audience forms with an interest in a particular kind of fiction that a genre solidifies, with it's own special rules and boundaries. So the question isn't so much "What makes Steampunk Steampunk" but "Why do people like it?"
So why do we like steampunk?
- Aesthetically, it's pleasing. Steam implies raw motive power, workings of pipes and engines give you something concrete and solid, but at the same time, it calls to mind a time when we were (at least superficially) more civilized. Steampunk feeds the imagination with networks of pipes, plumes of smoke, and chugging engines. It's unfamiliar enough that it catches the eye, but nonetheless easy to understand, which makes steampunk an interesting element in design, fashion, architecture, etc.
- Ethically, it's clean. Steam is clean, there's no pollution (necessarily) or noxious chemicals being poured into the atmosphere. Steam is guiltless. This may differ from setting to setting, as some steampunk does involve coal engines, but even then, it at least distances the audience and author from ideas of oil engines and global warming.
- Technologically, it's at one of the sweet spots for fiction, as far as allowing for possibilities. By this, I mean that steampunk can allow for all or most of the tropes of pulp science fiction, but it can also allow for all or most of the tropes of the wild west, industrial age robber barons, pirates, military and fantasy. There can be golems/robots and fights atop a train. It allows for a setting where guns exist, but they're not yet so advanced that you can't have a swordfight, not so crude that they'll misfire or wildly miss half the time.
- Steampunk rewards intelligent characters. Scientists, innovators, detectives, scoundrels, the general trappings of Steampunk society gives these people the tools they need to thrive, while keeping things dynamic and dangerous. Any of these characters can wind up on a steam powered airship that is venting too much steam. Any of them likely has the tools to deal somehow. Steampunk generally doesn't cater to thugs and brutes... which we, as a generally geeky audience, find appealing.
- It's romantic. Steampunk in general revolves around some of the most basic human dreams. It makes it possible to take to the air (airships, flying cities). It appeals to the person who dreams of coming up with a great idea that can change the world. Getting from one place to another isn't necessarily trivial, but it's always doable... so it feeds our desire for exploration and adventure.
At least, that's how I see it, from my limited exposure to the genre.