Why doesn't a shutter speed of ¹⁄₂₅₀th freeze motion when a flash of that duration does?

This question connects with my previous question: Why do cameras that film with high FPS rates cost a lot?

I just read that a normal flash illuminates a scene within a 1/250th of a second. (A flash would keep the scene illuminated for a 1/250th of a second, right? And it's the flash's sync speed, right?) So, is just a 1/250th of illumination enough for freezing motion in a dark room? And(then) why can't we use a 1/250th of shutter speed to freeze motion in a well lit place?