The only real difference that I can think of is the 'of' is used mostly as a possessive word between a part and a whole for example, "The sleeve of my shirt." Whereas the word 'from' is used a a preposition describing the origin of something for example, "This car is from Korea."

Hope that helps!

Jake's answer is correct.   There are also many more meanings for each of them.  I recommend going to dictionary.com and looking up both of those words.  

Dictionary.com reminded me of examples of "from" (that I forget about but use everyday):

1.  how it is "used to express removal or separation, as in space, time, or order".  The ocean is 2 hours away from here.  We are scheduled to Skype 90 minutes from now.  You never know what the weather is going to be like from one day to the next.

2.  "used to indicate cause or reason":  From what I can see, it is getting ready to rain.

(and there are 5 others ways to use "from"  and several for "of")

These 2 words are used frequently in both our spoken English and written English and it is important to learn, understand and utilize them correctly if you are going to communicate well in English. :)