Samantha,

Here is a dictionary definition:

FATE implies an inevitable and usually an adverse outcome <the fate of the submarine is unknown>. Something bad will happen for sure!

DESTINY implies something foreordained and often suggests a great or noble course or end <the country's destiny to be a model of liberty to the world>.

http://www.merriam-webster.com/dictionary/fate

Also, each word is often used in special expressions. They are popular in fantasy movies because fantasy movies talk about destiny and fate a lot.

You are destined for greatness! (You will become someone very important in the future)
Now, you will meet your fate! Now, you will meet your doom! - Now, you will die. (The villian, the bad guy says this.)
Your fate is sealed. There is nothing you can do to change the outcome of your life.

In addition to the definitions, when I hear the word destiny, I have a positive reaction. :-)
When I hear the word fate, I have a negative reaction. :-(