All-in-all, the Core Rulebook will give you more in depth rules. It gives you a wide variety of situations and rules, has more adversaries, has more options, allows character creation, and lets you delve deeper into the Star Wars universe we all know and love.
From what I've seen as a player of the Beginner Box in Edge of the Empire, and also by flipping through the Beginner Box rules for one of my own hodge-podge games, the Core Rulebook is just better. Better condition from hardcover, lots of artwork, explanations for a large varieties of areas and circumstances, and all kinds of other good stuffs...
I, myself, first doubted getting the Core Rulebook, but just walking in the store and seeing the pure size of it was good enough for me. So yeah, there's a big difference, and you should definitely get both (Beginner Box to learn to play and get everyone familiar with the rules, Core Rulebook to continue from there).
"The five guards could have been a coincidence. An entire cargo hold of aliens, sprawled out, burnt, in neat lines, with horror still struck on their faces and children still clutching in death to their mothers and men standing valiantly but to no use in protection of their wives was not a coincidence. It was a massacre."
-The Horrors of the Empire http://community.fan...re/#entry989920