I think the question may be:

What significant difference does it make if we put a woman (or woman of color) on the moon first vs putting another white man on the moon. [with modern technology].

I could think of a few positive reasons to do this, but it shouldn't be the main driving force of competition.

No scientific difference. It's political. A president can trumpet it as a great achievement for humanity. Helps with getting funding & public support, yada yada.

It feels like the value in political virtue signaling is quite past its peak, in fact I think there's something of a negative value to it in a lot of important circles.