They both primarily aimed at photo editing and enhancement. But Photoshop goes so much further having many more bells and whistle. I am not sure how much you understand of the terminology, but some of the features in Photoshop that you will not find in Elements are the pen tool, curves adjustment, channel mixing and CYMK support.
The choice (apart from budget) comes down to how much photo manipulation you need to do. Standard stuff like cropping, colour adjustment and perspective is handled by Elements. I use Elements and it is fine for my needs. It is also good at erasing minor blemishes. However, if you intend to get involved professionally or as a major hobby the full version is the way to go.