Annika plays on the same course when she plays with the men on the PGA Tour. The LPGA Tour golf course is set easier than the PGA Tour course.She is playing the same game..same tees..same rules.
I am not a woman-hater ( ) but the reason why many PGA golfers complain about women playing on the PGA Tour is because they take a place of a PGA player. I think that if the women play the Monday qualifier or if they get into the PGA Tour via the PGA Qualifying Tournament then there will be less resistance.
Male players in many sports have complained about women getting into the men's team based on criteria other than merit. Annika is dominating the LPGA Tour but I am not sure that she can play better than the top 125 PGA Tour players if she has to go out and play on the PGA Tour week after week. Keep in mind that the women players enter PGA tournaments where the golf courses suit their game.
One can argue that Annika's presence on the PGA Tour event via a sponsor's exemption brings more paying people to the tournament/higher TV ratings and hence the Tour makes more money, and some of it may end up with the players. Some male players are afraid to lose (based on # of strokes per round) to a woman whether the female golfer makes the cut or not.