Yes, the trial version of lightroom is identical to the full version.
I used it based on the advice from this board, and now I can't live without it.
It is very different than photoshop elements. While it can do many of the same things, it does them differently. It is a bit difficult to explain, but I'd put it like this:
Elements is great for photo EDITING. Making structural changes to the photos themselves. Adding or subtracting elements from the photos. You want to take your picture of Mickey Mouse, and put him on top of Big Ben, then Elements is the way to go.
Lightroom is really more akin to the old fashioned dark room. It is not for heavy duty photo editing. It is great at photo organization, and is great for photo adjustments. A photo can be enhanced significantly just by adjusting the lighting, shadows and highlights. Lightroom is also very powerful at noise reduction. And it can serve your very basic editing needs -- it will crop pictures, it can remove spots/blemishes, remove red eye.
Now while you probably can make similar lighting adjustments on Elements, on Lightroom it is incredibly intuitive and easy.
If you have an iphone or ipad, there is a photographer who has a cheap app tutorial of how to use lightroom. Lots of videos taking you through it. It can lead to rather amazing results.
My own personal video editing suite has become -- In order of how often used:
1. Lightroom 4.2. I am now primarily shooting in raw, and giving every picture a little tweak.
2. Photomatix -- I enjoy occasional HDR. There is a plugin from Lightroom, so Lightroom exports the pics right into Photomatix.
3. PSP5Pro, with Topax an Nik plugins. Used very rarely. Similar to Elements. Allows me to sometimes do more advanced editing, but rarely do I have much use for it. Using the filters, can sometimes add a bit of extra flavor to a pic.
Truthfully, if I wasn't going to ever do HDR, I could live with just Lightroom.