Down is the opposite of up and it indicates movement towards the earth. Below and under are similar, they indicate a place in reference to another object that is above. Personally, I see under being more related to the above object. The plates are under the cups. The cups are important in this sentence. When you see them, you will know where the plates are. I think below is used more when it is the person that is important. If I say "no, it's below the mountain" or "they're below the sink" I am asking the person to redirect their attention. But, in reality, they are interchangeable.

Here is the link for your reference https://www.phrasemix.com/answers/whats-the-difference-between-under-below-beneath-and-underne