Yes, there is a big difference between the two: "je suis faim" makes no sense in French. 

What you were taught in high school most likely didn't *suggest* that the proper way to say "I'm hungry" is "j'ai faim"; it probably stated it. And it was right. 

You could look in a dictionary and see that "faim" is literally "hunger". So yes, in French you "have hunger". It's a fixed expression that you can't really mess around with. If you want to say that you are very hungry, then you'll say "j'ai *très* faim". But if you want to use *être*, you'll have to use something other than "faim".

You might be interested in this question about [hunger-related expressions][1] (in French).


  [1]: http://french.stackexchange.com/q/5696/217