Unfortunately, this is exactly you should require a contract and upfront deposit of some percentage of the final bill (usually 40-50%). You probably don't have much recourse legally unless you can prove that an email chain at the beginning of the project is detailed enough to constitute a contract.
Another thing that you can do is to take the client's new requirement and redo that part of the project. If you and the client were in constant communication throughout and they sprung the no-longer-a-requirement card, that's one thing. However, from the post, it sounds like the communication had been limited. Requirements do change, and sometimes you have to keep in constant contact with your client to make sure the project is still going in the right direction.
Frankly, your best bet is to chalk it up to a life lesson, spend some time to create a contract boilerplate (Contract Killer 3 is a sample starting point), and use it with your next project and next client.