What is a good example to illustrate the difference between isomorphic and homomorphic representations?
You are probably talking about homomorphisms from logic, i.e. homomorphisms that just preserve your functions and relations: https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

I'm a bit suspicious about why your instructor bothered to give you these concepts in the intro course. I also suspect you just need to understand a difference between injective and bijective functions (for this is what the difference between a homomorphism and isomorphism is in the logic world, ignoring all the stuff that deals with preserving structures).


Injective function. You have a set of shirts. You represent the shirts by their colours. Many shirts have the same colour. Some colours don't have any shirts mapped into them. If you are given just a colour, you cannot recover the knowledge of what shirt it was.
Bijective function. You have a set of books. You represent them by the ISBN number. Every book gets and ISBN number. Every ISBN number is associated with some unique book. You can go back and forth between these representations.