Watchmen is basically the story of superhero vigilantism gone wrong. The graphic novel borrowed visual and storytelling elements from superhero comics and gave them a twist. In order to make a movie, they had to pare down the story to just some essential events. The graphic novel has a lot more to it than the movie, but the movie does a good job of, well, being a movie.
Summary of the Plot of Both
In the 40's, a group of costumed vigilantes called "The Minute Men" became valuable allies to the cops and celebrated crime-fighting heroes. Some die, some retire. In the 60's, their successors, and some members of the original team, became "The Watchmen" in the same vein. But after Vietnam, when the war was ended by the use of the actually super-human member of the team, Doctor Manhattan, the group (and all such costumed crime-fighting) became illegal. This means some of the group who had been following in the older members' footsteps, like Laurie, who had taken her mother's place as Silk Spectre, were left cut out of the business they'd been learning from their elders. Another blow to The Watchmen is struck when Ed, whose alias was "The Comedian" is found dead. Old teammate Rorschach, who wears a white, ink-blotted mask that is forever changing, investigates. Rorschach thinks that someone is trying to kill all the former Watchmen. So he goes to interrogate former team members for further information about who might be doing so and why.
At first, we start off focusing on Rorschach's perspective. Then, it's about Laurie Jupiter. She's the daughter of ex-Watchman Sally Jupiter, who's now retired. She's living with Jon, aka Dr. Manhattan, who has bizarre powers from a nuclear accident, but is kept by the military as basically a human reserved weapon. So since she doesn't get to get out much and Jon, working as a researcher, isn't the most fun company, she ends up romantically involved with Dan, another former Watchman who was known as the Night Owl.
Then we get into Jon's perspective. He sees past, present, and future, simultaneously, but that can make him prone to being detached and aloof in the present. At first, he is reluctant to get involved in any conflict, but when he discovers that the bad guys have tampered with his mind, then he's involved. And the bad guys better look out. Who is trying to hurt Dr. Manhattan and why? Find out by watching the movie, or better yet, read the comic.
Comparison: Comic vs. Movie
Like most book to film adaptations, Watchmen had to cut out quite a bit of the source material to make a movie short enough to please audiences. Between chapters in the graphic novel, there are faux documents such as letters and journal entries, and these are long and complex, too much so to have a place in a movie format, but I found them interesting. The book is long and wordy, but it's chock full of good, quotable lines, many of which made it into the movie. But I'd definitely recommend checking out the comic book if all you saw is the movie, because you're missing a lot of good stuff. If you're a fan of the original comic book and not sure if you should see the movie, it's a faithful enough adaptation, but you might find it to be too watered down in comparison. I think the book "wins", but the movie is still enjoyable.
Basically, if you want to be entertained and intellectually stimulated at the same time, Watchmen is great in either form. Since Watchmen is a "comic book" that elevates the medium and shows that a comic book can be a great work of literature, the movie reflects that sophistication. But if you truly want to explore the deeper philosophical and political messages of the work, I recommend reading the source material, the comic book. The movie is just the plot, but the comic book is the plot with volumes of depth that could not have made it into the film adaptation.