The MAT is a route for teacher certification. I didn't decide to become a teacher, or have interest in teaching, until many years after my undergrad career. I had the option of a route to certification without receiving my Master's, but I knew I wanted to go further anyhow. The Masters of Arts in Teaching degree not only focuses of content, but also applied research and teacher candidate teaching under a cooperating teacher of record.
The M.Ed. is really for those who knew during their undergrad time they wanted to become teachers, and have gone through student teaching and met all the requirements for certification. This degree is also good for those who are more interested in policy or curriculum, and do not want to become teachers. However, I will say, that even one year of teaching will give you a greater insight of policy reform and curriculum needs. The theory of teaching is so fascinating and I am a huge advocate for anyone interested in our education system, but some things are brought out in the classroom that theory will never truly show. Even if you never want to teach, become friends with a teacher and visit. Often. And, ask if you can help out sometime. It is an experience you will never forget and it will solidify what you know or give you a greater perspective.