The difference between different colored eggs is.... the COLOR!
Seriously though, it's a very common misconception that different colored eggs have different properties, when in actuality they're all the same. Its what the chickens eat that changes the make-up of the egg (how healthy, how it tastes). For example, free-ranging chickens tend to have more omega-3 fatty acids and have a much darker yolk which tastes much better (I think).
Here's more information than you probably want on egg color:
Egg shells come in 2 base colors, either white or blue.
Most breeds have a white base, but some also have blue (EEs, Ameraucanas, etc.)
To get a brown shell, the hens put a brown coating on the egg (white base). To get a green egg, the hen puts a brown coating on the blue base. The color egg a hen lays is determined by her genetics.
A broody hen is one who wants to raise a "brood" of chicks. She wants to sit on her nest until her eggs hatch (21 days) and then will typically mother her chicks.