Fame
Cool name, cooler fashion sense (source).
The time of famous nuns and monks ended around the same time that the Renaissance began. Pilgrims don't flock to see modern-day nuns and monks—they visit the tombs of those from the past, like Saints Augustine, Patrick, and Hildegard of Bingen.
Occasionally, someone like Mother Teresa comes along. She's world-famous for her incredible charity and kindness—but you can bet that she didn't do all those nice things for the newspaper headlines. For the most part, nuns and monks spend their prayerful lives outside of the spotlight.
Sure, some nuns are more active in their communities, meeting with other members of the congregation and occasionally heads of state. But the paparazzi aren't stalking them on their run to the grocery store for cleaning supplies.