For all you geeks out there here is a great description of the new HDR+ enhanced mode:
HDR+ ... first, ignore the HDR, because that's not what it is. Google got a whole bunch of imaging ubernerds from all over the land, especially guys who worked in astronomical photography, and had them apply principles used with telescopes... devices trying to get GOOD pictures without a lot of light to play with. And artificial image processing is completely useless to an astronomer. They'd rather a blurry, noisy mess than something processed through an an algorithm, which is another astronomer's best guess what that looks like. So they came up with clever ways to improve an image without screwing with it. HDR+ is sort of a cousin of those techniques.
So what does it do? When you take a picture, the Google Camera takes a series of quick exposures with the exact same settings... 3, 5, as many as 10... and it then goes in and picks out the one that it deems to be the 'best'... it could be the one with the least blur, the one where your kid has his eyes open and looking at the camera, etc. It makes that the 'base' picture.
Then it takes all the other exposures and starts comparing the images, pixel by pixel. Simplified way of looking at it is it averages out each pixel across all the shots. Since noise is random, this will filter out a LOT of the noise and you end up with a value that is pretty much true. It also then goes in and analyzes the image to determine the proper intensity of the pixel (how bright it is). It'll then go through and stitch all those together to form the final image. It might then do a little bit of artificial noise reduction and sharpening, but not much... Google tries to use the image itself to improve its quality rather than an equation.
OK.. that's HDR+... Now, 'On' vs 'Enhanced'. It's all about HOW it takes those images.
In 'On' mode, the camera app is continuously taking exposures and keeping them in a buffer. When you hit the shutter button, it then goes in and captures the last few shots in that buffer.. more if its dark, less if it's bright (the brighter a picture, the better chance it won't need a lot of work done on it). Then it processes them.
In 'Enhanced', it goes a little deeper. Rather than using the continuous steam, it stops that buffer. It will then go an try to set the exposure on the brightest part of the image while also increasing the light sensitivity (ISO) quite high. Now, for most cameras, that means a dark and noisy shot. Why take dark noisy pictures? Well, HDR+ is pretty clever. It can go an in clean up the noise in those dark regions and use the information to boost the lowlights. So if you are taking a picture of someone with bright clouds behind them, no only will you be able to see their faces, the clouds behind won't be blown out.
Now, since "Enhanced" takes more grunt work, you lose that instant shutter speed of the default mode. But what you gain is the ability to take a good picture in an extremely challenging situation... usually where you have a wide range of lighting, or if you are trying to take a really dark or night shot.