I have seen Godzilla, the new movie, but am not allowed to talk about it in any detail until next week. Godzilla: Awakening, on the other hand, comes out Wednesday, and I can talk about that, because I have it in hand.
In a nutshell: if you want to be spoiled for many significant plot points, they can be found in this hardcover.
If you have seen the movie, on the other hand, it fills in many details merely alluded to, like the origin of the neologism MUTO, and the purpose of the organization called Monarch. Essentially, it's the story of Ken Watanabe's character, Dr. Serizawa - but as he and his knowledge don't fully reveal themselves for a while on film, this gives you arguably a premature primer on what is really happening in the world before most of our onscreen protagonists know it.
Also, as you may suspect, the reveal of Godzilla in the comic is considerably more generous than special effects budgets allow onscreen. By now you know what the new version looks like anyway, but seeing him this much on the page beforehand may make it less mysterious in a theater.
The secondary monster in the comic, however, is NOT in the movie. It's a bit of a riff on both the the Smog Monster and Ygramul the Many from The Neverending Story (book, not movie) - a Brood-like kaiju made up of swarming composite parts. When it finally fights Godzilla - who is named Gojira throughout the book - it's on an island called Moansta. Get it? "Moansta Island"? That's a nice little in-joke, as is a throwaway visual gag suggesting King Kong, or something very much resembling him, also existed in this world at one time.
Artwork by Eric Battle, Yvel Guichet, Alan Quah and Lee Loughridge is solid throughout; the story, by Max and Greg Borenstein is necessarily episodic (it is filling in bits and pieces, after all), but coherent enough to work as a whole, especially if you've already seen the movie and want to know more.
If you haven't? You have to ask yourself how much you want to know right now.
Godzilla: Awakening comes out May 7th from Legendary Comics.