A snood implies more casual (like a bandana print, or one that has D & G embossed on it). A tichel is more dressy (like Hermes or Channel).
As one who has attended many PTAs, when you come in dressed like you mean business, people take you seriously. (this applies to both parents and staff)
When you come dresses in whatever suites your fancy, you’re placing yourself at risk for being underestimated. Its hard enough to get noticed in today’s anon world. Take advantage of every edge you can.
And kids? Oooh boy, do they pick up on these things. Many of our friends are in chinuch, and the ones that make an impressive stance (clothes, grooming, ect) have a much easier time doing their jobs.. becuase they are only fighting one battle, and not trying to overcome perception at the same time.
Sorry to admit it, but we live in an exterior world.