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", like "affamé", which is an adjective derived from faim.
You might be interested in this question about hunger-related expressions (in French).
