Watchmen is a important comic, quite possibly one of the most important comics ever made. It introduced a whole host of themes which other writers and artists would pick up on in the following years. What do these extreme personalities we call "superheroes" do in their personal time? Should superheroes get involved in global politics? Do cosmic-powered blue guys really need pants?
Certainly, it added mature themes to comics, but more importantly, I don't think any other comic series reached the level of artistic merit that Alan Moore and Dave Gibbons achieved.
One of the most amazing things about Watchmen is it's verisimilitude. Each chapter ends with a fake magazine article, newspaper clipping, police report, etc, which gives you more insight into the setting and the characters. These show you just how much thought Moore put into the piece while crafting it. For example, in a world where superheroics are an everyday occurence, what kind of comics do those people read? It's a bit of detail that often gets overlooked, but Moore not only answers that question, he makes it an integral part of the story.
How important is the Watchmen? Very. It's one of the few comics you can read and not get called a "nerd". Also, The Simpsons made a joke about it, so it's even further entrenched in the popular culture.
Is Watchmen still relevant? That's a good question. The Cold War is over; there are still nuclear threats, like Iran, but the underlying idealogical concerns behind those threats are much different. Still, as Watchmen illustrates, it's not so much the threat as our reaction to it. The gamut of idealogies is shown within its pages, from Rorshach's Ditkoesque conservatism to Ozymandias' frighteningly practical liberalism. As long as people continue to make the same kinds of mistakes, Watchmen will still be relevant.