Throw away means to discard or waste something, especially something that can no longer be used or is ruined. Usually you throw away something into the garbage bin.
Ex: I threw away my old shoes after our hiking trip in the mountains because they were dirty and had holes in them.
Ex: Please throw away those scraps of paper once you have finished the project. The scraps are so small; we can't use them for anything else.
You can also throw away an opportunity, if you waste, give up, or ruin a chance to do something.
Ex: Daniel was a very talented young actor, but he threw away all his talent and wasted it on drugs and alcohol.
Dump can refer to the site for depositing waste or the act of emptying or disposing a large collection of waste.
Ex: When are you going to take all that rubbish to the dump? It is beginning to make your whole garage smell, so you need to get rid of it.
Ex: The woman dumped the clothes from her washing basket onto the bed and quickly ran to answer the door.
Ex: Peter has been sad and unsocial lately because his girlfriend of six months just dumped him. He really loved her, but she didn't want to move closer to him.