I'm no expert, but from what I've learned about yoga, ashtanga yoga is a system that literally means "eight limbed yoga." Each limb deals with a different aspect of ashtanga yoga such as morality, purification, breathing, etc. Whereas vinyasa is more like a dance that flows from one movement into another and focuses on synchronized breathing with each ashtanga movement.
Upward movements incorporate inhaling and downward movements incorporate exhaling. The breathing is critical and when correctly synchronized with each movement, the results can be improved cardio-vascular function, increased muscle strength and flexibility, and reduced stress.