浴室 emphasizes the function as bathroom, 洗手间 emphasizes the function as toilet. 浴室 and 洗手间 are usually combined at home, so they are interchangeable when you are at home. While in public places, you have to say 洗手间 if you want to use the toilet.

浴室 = bathroom (the room with shower and / or bathtub) 洗手间 usually means "toilet" in public places, like restaurants and bars

浴室 is your bathroom at home where you can take a shower/bath, but a toilet is not necessary for it though usually we have a toilet in our bathrooms. 厕所/卫生间 can refer to a private bathroom with a toilet. 洗手间/卫生间can also refer to a public restroom. Obviously you can't take a shower/bath there. In addition,公共澡堂=public bathroom；上厕所=use the toilet(private or public)（more politely as去一下洗手间）

浴室 = bathroom with/without wc 洗手间 = 厕所

解释的很好！