I would argue that we are more defined by the small, everyday challenges. This is because they are much more common and constant and therefore make up a bigger part of our lives.
Great challenges come along very rarely. They do not make up a large part of what happens to us so they should not be what defines us. For example, if I react splendidly to a grave family crisis but am irresponsible and unpleasant on a daily basis for the rest of the time, which carries more weight? To me, it is the daily actions that I take that would define the way my family sees me.
Since small challenges are the things that make up the vast majority of our lives, they are the things that define us to a much greater extent than greater challenges do.