Actually, the reason I find most convincing (and perhaps the most obvious) is the issue of practicality, and not some urban legend about personal rivalry. Mathematics is not a science; it is much more abstract than science, and as a (general) rule, the more abstract a concept is, the less practical it is as well (when was the last time quantum physics was put to something useful?
Entire branches of mathematics have been discovered and researched that have no relavence whatsoever to the real world (solving Fermat's last theorem comes to mind). Alfred Nobel created the prize with the intention of rewarding those who, through their actions, had a positive influence on the world. In other words, if one wished to win a prize, he would have to discover something that was more than just intellectually tittilating.