There's loads of differences between lightroom and Elements. Lightroom is first and foremost a photo management system with some excellent editing capabilities built in. Elements is first and foremost a photo editing tool which has some catalogueing capabilities.
In addition to the management capabilities in Lightroom, it provides workflow capabilities, where you can edit/manipulate groups of photos at a time, and do all this either directly within Lightroom, or do things automatically as you import into Lightroom. The editing capabilities do not rival Photoshop, but are absolutely more than enough for a beginner/advanced hobbyist. You can link lightroom and elements or Photoshop so you if you feel you want to do further editing in an 'external' package, you can do that from within Lightroom.
Further, Lightroom edits are 'non-destructive' which means you can backtrack through the edits to see what individual changes does to the photos. Elements changes the picture and doesn't allow for 'backtracking'. and also edits a single photo at a time.
I use Lightroom to manage my pictures and do the bulk of my editing. I switch to elements to do some extra edits on a single picture. (eg. selective coloring comes to mind.)
I would recommend using Lightroom over Elements, and add Elements when/if you need to. Lightroom handles 99% of the editing I need to do.