The main answer is water resistance. It's not the whole story, not by a long shot, but it's the primary significant factor. True diving watches must meet the international ISO 6425 standard, part of which sets the minimum 100 m. requirement mentioned above (dive watches can often dive much deeper - that's just the bare minimum). If a watch can meet this exacting standard, it can be engraved with DIVER, which is how you'll be able to distinguish the real deal from a mere lookalike.
People have been trying to waterproof watches for ages (indeed, since timekeeping was especially important for shipboard navigation, and since navigable ships are, by definition, surrounded by water, it was to some extent a top priority). Dive watches are the outgrowth of these previous efforts. They expanded on the needs of naval navigators and other people operating in and around water with the goal of creating a watch that could truly be used under water. Many early models were produced by or for militaries.
And then came Rolex. 1954 brought not just the first scuba apparatus but also the first Submariner - two truly complementary innovations. While scuba gear meant that divers could go deeper than ever, true dive watches made attaining those depths safer than ever.
Of course, diving watches don't just have to keep ticking - there are a slew of other requirements to be met to keep divers safe. They must be legible in low-light or no-light situations. They must include a means of timing the dive and the crucial surface ascent - if you don't get back to the surface in time, it doesn't matter how reliable the watch is.
Dive watches must resist more than just water - they also face magnetic forces, shock and rough impacts, and chemicals. The strap and assembly must be tough and durable. And the diver must know that his or her dive watch really is working in all conditions - and also know when it's about to stop working.
If you're interested in diving watches, and not just for show, then you'll need to carefully maintain them as well. Just because they're built to be tough and reliable doesn't mean that they can simply be ignored. Dive watches are precision tools and like all tools must be cared for.
You'll need (and want) to have them serviced and inspected every two to three years (or yearly, if you can manage it). The seals of dive watches must be replaced from time to time. Every use, you'll need to wash them in fresh water to remove salt which may begin to corrode any exposed metal. And be sure to take care to avoid too much exposure to harsh chemicals. They might be ISO certified, but they're not invincible!