After checking the dictionary, they are practically the same.
However, 寝る suggests sleeping while lying down (or even being an euphemism for sex) while 眠る is for sleeping but not necessarily lying down (also a euphemism for someone who has died). Hope this helps.
if you understand some Chinese, this gonna be easy to figure out. 寝 this kanji in Chinese means 就寝，睡觉 that is to go to bad or sleep with or lie down. as for kanji 眠 means 睡眠，入睡，永眠 that is to say to sleep (not necessarily lying down) or to die