When you put down your water glass, you simply place it on a surface. However, when you put down your math teacher, you criticize her harshly.
You can put down the grocery bags on the kitchen counter, or put down a puppy when it starts squirming. You can also put something down on paper, or write it out; and you can put down, or pay, a deposit on a new apartment. When you put down a person, you denounce or find fault with them, usually verbally.