I understand it when in an athletic competition, such as the Olympic games, one struggles to be the best in his club or country or the world. What he wants is to be champion; there is an aspect of glory in this; and seeking glory is a genuinely human vanity.
But why would anyone seek to be equal to anybody else? What does the motivation to equality consist in? What good am I gaining when I find myself equal in whatever aspect to X?
Both victory and equality are "relational" goods. For example, I am a champion when I am better than Smith, Jones, and everyone else in some sports or other. Whether I achieve glory may depend on where other people are relative to me and in particular whether I am superior to them. Similarly, I am equal only in relation to others.
And yet the search for victory in competition seems perfectly reasonable to me, and the search for equality does not.