The vegan diet is necessarily strict. The diet is a belief system put into living practice. The true vegan diet is full of 'may nots' and these include anything 'animal' that was ever living, could ever live or any piece of which might be used in anything they consume or use. Basically, don't even think about it if ever it wiggled!
So where the vegetarian cannot eat the chicken, but can eat its eggs, the vegan diet forbids eggs. The vegan diet also forbids anything animal or any animal by-product, which then extends to all dairy products, honey, hides and feathers, hair/fur including wool and the usage of any product in which animal products are an ingredient such as soaps and cosmetics. That's a pretty strictly defined set of living parameters!
Why would one chose the vegan diet? Well, veganism is chosen for health, environmental and ethical reasons, depending on the individual.
For instance, one on a vegan diet might ask you to consider the "modern" chicken coop. It bears scant resemblance to the farmyard of old. There are no chickens scratching at the earth trying to find a hidden seed. Instead, there are thousands and thousands of chickens that have arrived and departed this earth whose feet never touched the ground!
A vegan diet acknowledges that, with effort, we can free animals from slaughter and unnatural lives. A plant-based diet is far better for humans and we no longer need to hunt semi-sentient life forms to eat well.