To my knowledge from what I learned in my first and second year of Japanese 寝る is more for like going to lie down or to rest or sleep while lying down. 眠る is more like actual act of sleeping. I rarely have used 寝る for actually saying I want to sleep, more for wanting to lie down and rest. Of course 寝る also depending on context means to sleep with someone (in a sexual way). But I've mainly seen it used for to lie down. 眠る is straight up to sleep. It also doesn't automatically mean you're sleeping lying down. You could be sitting in a chair or in a car, whatever. It can also be used for to die, but I don't think I've seen it used in that way from things I've read. 眠る can also be used for saying someone closed their eyes.

Hope this helps you !