What's the difference between yes and yeah?

Yes and yeah mean the same thing.  The difference is that "yes" is correct and more formal.  "Yeah" is casual and may sound disrespectful in formal situation.  For instance, in Court you would not say "yeah," you would say "yes."  In a classroom also, "yeah" is not quite right.  
"Yeah" is okay in most ordinary conversations.
I use "yeah" in Yahoo!Answers, because it is a casual and conversational forum.  

Yes is formal and always correct if in any doubt.  Yeah is informal and can sometimes be considered disrespectful if the situation is formal or serious.  I would never have used yeah with a teacher, for example.  It can also express a grudging or sarcastic "yes" depending on how it's pronounced.  

'Yes' is the correct word.  It's polite, respectful and formal.
'Yeah' is slang, casual and a lazy way of saying 'yes'.

Yes sounds intelligent. 
Yeah sounds like a defiant or ignorant child.

"Yes" is the root word and is formal.

Yeah is colloquial.  

"Yes" is a straight answer and "yeah" is a grudging or quick answer.

Thanks for spending the effort to learn English. You are correct that they both mean the same thing. As for usage, "yeah" is a casual response that would not be used in a formal publication, college theses or similar paper. "Yes"
 is a suitable word to use for the same thing and is suitable in any context. Hope this helps.