Some good answers are already given in this thread, I wish only to hint at the whole class of metaphors which can be used as a quick and cheap answer:
The difference between mathematics and mathematics education is the same as
- between religion and religious education
- between ***** and ***** education (you may wish to continue the list using this pattern)
Answering this question, it is very easy to switch into cynicism — one of the responses in this thread, from a former student, is already highly critical of mathematics education. I see myself as working professionally in mathematics education, and it is painful for me to see popular sayings as this one, from American popular culture:
Those who can, do. Those who can’t, teach. Those who can’t teach, teach teachers.
— but I have to live with it. We have to accept that this is a popular attitude to our profession.