Audio quality is accurate and responsive in both types. Which is best is a matter of personal preference and listening situation.
Open-back headphones
Open-back (open air) headphones use some external airspace for a more natural, life-like sound. You can control the volume with precision. They can leak some sound at high volumes.
Closed-back headphones
Closed-back headphones seal around the ear to isolate you from your environment and prevent sound from leaking out. Great for focused listening or if you don't want to disturb others.