Men's watches are bigger, prefer metal over other fabrics for the strap, and are much more flashy, meant to catch your eyes instead of complimenting your overall clothing style. Women's watches are smaller in terms of wrist size and the actual watch size, use less metal, and are more subtle. You can wear either but the two genders are undeniably different so it makes sense that they have gender specific options.
"Watch for men? How is it different?" I was answering that. You're free to wear whatever you want but the point still stands, men and women typically wear different watches because they are just different so it is not completely out of the question for your mom to deny you that watch. Besides, watches are expensive and she might've been looking for a reason not to buy you it.