Very good topic for debate.
In an ideal world, this would not at all be needed. In the real world as it exists now, perhaps this is still a necessary evil.
As far as sports go, it seems that if women and men were to compete, there would be many sports in which men would be completely dominant, and maybe a few in which women were dominant. This would be discouraging to, for example, women who want to play professional basketball. Whether this domination is due to genuine biological differences between the sexes (and, of course, we're only talking about average differences, for what little they may be worth) or to the different way in which men and women are raised in society is a subject for another debate.
It's harder to defend things like the Oscars. Why not Best Performance? I doubt whether one sex or the other would dominate things. Of course, this is such a tradition that I doubt it will change in the near future, if ever.
One area in which the sexes have been separated for competition is chess. If there were not women's championships, men would be utterly dominant. Is this because chess at a championship level makes great physical demands, like sports? Is it because men have some biological advantage in the brain for chess? (Again, I'm just talking about averages, not individuals.) Or is it because of the way that men and women are raised in society, with particular emphasis on their level of aggression? Again, interesting topics for another debate.
Thank you for giving me a lot to think about!