The new corebook has been revised, updated and expanded across the board. There is intro fiction to set the tone of the book, but even not counting the intro fiction the book is about 20,000 words longer than it was before, and it's all new material.
The most obvious change is that all errata from the first print has been fully incorporated into the new printing. Naturally, as new issues crop up we're recording them and will continue to make corrections in the future. We've also gone back to the 12/18 CB Prime/Secondary split for SIEGE checks, to keep it 100% in line with C&C, but I put in a sidebar in the GM section for those who want to stick with the 15/+5 method of the first print, so that option is still there.
As far as brand new stuff, some of the character classes (the Pugilist, Gadgeteer) have been overhauled similar to the way the Barbarian and Illusionist were in the C&C later printings. There are a lot more new Generic Class Abilities. There is an entire section dealing with Secret Societies, how to build them and use them in your game, as well as several sample societies. There are (optional) guidelines on making magic a corrupting force tied to the sanity of the Arcanist. The monster section isn't as big, but has been augmented with a Rogue's Gallery including notable figures, NPCs and pre-generated player characters. A lot of stuff has just generally been clarified and better explained overall.
The original AA, while I'm quite proud of it, was my first ever core rulebook for a large publisher. After over a decade in the industry I was thrilled that Steve and Co. gave me a shot at it, and I truly love the game. This revised printing really highlights its strengths as a game, I think, and sees it reach its full potential.
Hope that helps! If there's any specific questions you have I'm more than happy to answer.
Oh! One more thing: while the new Manual of Monsters DOES include the complete bestiary from the first printing, it also includes over two dozen, never-before-seen monsters to add to your game so you are NOT just re-buying stuff you've already got with that one. We're not about making fans do that! And if you decide NOT to update the corebook, the new Companion and Monster book will still be 100% compatible with the first print, though the page references won't match up so you'll have to hunt and peck where that's concerned