1. 화장실 means powder room. 화장 (make-up) + 실 (room) And it refers a toilet in common whether it's for public or not. 2. 목욕실 literally means a bath room. 목욕 (bath) + 실 (room) And we don't use it much and when the time comes to use this word, it usually means a private bathroom like installed in residential houses. 3. In addition to these words there is a word for a public bathroom in Korean. And it is 목욕탕. 목욕 (bath) + 탕 (originally means "boiling" but in this case it means "hot tub" with boiling water in it.) It is a spa including sauna and other body care & relax facilities.

화장실 is a toilet, 목욕실 is a bathroom 화장실 refers to a public toilet, and also 화장실 refers to a private toilet and bathroom when both are installed in same room.