"Run down" can mean to chase and catch someone or something. It can mean to deplete, like "You are going to run down the battery. It can mean you are tired. "I am so run down." It can means to find information about something. "Run down the facts about the case." "Run out" can mean to deplete or use something up. "We are going to run out of sugar." It can mean to go get groceries or something else. "Run out and get some sugar." These are some uses.

You can "run down" another person with negative criticism. e.g. He consistently ran down his rival in front of the entire meeting. When you learn to use new phrasal verb, you need to learn which of the 4 structural categories they belong to, so that you construct sentences correctly with them. Confusingly, sometimes a phrasal verb can have more than one meaning and the different meanings are categorised differently.

It's best to consult a dictionary, or even better, a phrasal verbs dictionary. This is because there is more than one meaning for both words. If you do that, you could then come back here and ask a more specific question on anything you are still not sure about.

You have run out of places to hide! We ran out of milk! Did you just run down that hill? Don't run me down like that.

Keep in mind that there is a word "rundown" which has other meanings. http://www.thefreedictionary.com/rundown