In a formal sense, the main difference between teacher and student is the amout of education.To be a teacher, one must attend school for a certain number of years; take varied courses relating to one's discipline, as well as general courses; and earn either one, or two degrees, depending on the level at which one plans to teach. Some go on to that third level, the Doctorate, which allows them to teach at the graduate level.
All good teachers are always students in a very broad sense of the word, and sometimes teachers take classes where they are actually students of other teachers once more. All students are teachers in a broad sense as well, in that they offer new and fresh perspectives to the materials that their teachers are using in class.
The more we learn, the more we realize that there is too much for one person to learn in a lifetime.