Because neither believes the other is an unworthy worm. Egwene had to believe that of Mesaana before her mind snapped.

As for a direct TAR battle... Egwene, of course. She can already banish entire nightmares with barely a thought, like the Wise Ones. Perrin is still not at the point where he can do that.

People are taking entirely the wrong thing away from Perrin stopping Balefire. Like he said, it is only a weave. A weave used by someone with no skill at TAR, at that. That it was balefire and not a trickle of Air makes no difference. Both would be equally easy to stop.

At the moment, all Egwene needs to do is create a nightmare to trap Perrin, and she'd have him. Like what Moghedien did to her, only she's be better because she actually knows Perrin, so could come up with more realistic scenarios. Perrin needs a lot more practice before he's Egwene's equal in TAR. But it looks like he will indeed reach there some day.