Just wanted to clarify.. traditional HDR often uses what's called exposure bracketing. Three or more exposures are taken in a -1/0/+1 scheme and in post they kick parts that are either overexposed or underexposed.
HDR+ is not traditional HDR.
HDR+ uses what is called 'Lucky Imaging'. It's a technique used in astronomy to try to extract the best image possible from far too like light. The camera takes a series of quick, identical images (which in the Pixel is being captured whenever the camera is running).. same exposure, ISO, etc. It looks at the images and picks the 'best' and uses that as the base. It then goes pixel by pixel and compares that pixel across all of the exposures. This will reduce noise (which is random) while building information that can be used to fix the exposure later on. Once that is done, there may be some additional noise reduction and sharpening (HDR+ doesn't apply much of either though) and then it'll do some exposure compensation using the information gathered earlier.
Now, enhanced takes things a little bit further... Unlike the default HDR+, which utilizes the always on image stream, enhanced stops the stream. It does this so it can adjust the exposure, usually to better expose the brightest parts of the image. Then it takes more exposures than the default HDR+ mode. The time combining these two steps is the reason for that shutter delay. It'll then go and process like normal, but maybe goose the various processing parameters a bit to increase their effect.
Now, you might not always see a big difference between the two modes, it'll all depend on the conditions. But, in general, in scenes where there is a ton of dynamic range, enhanced will probably give you the best shot. But it's not necessary.
Here is a shot that would be damn near impossible with any other phone. I took this pick at dusk, and took a quick snap in default. In the full, it managed to expose both the brightly lit field and the darkened sky in the background.
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Now, here's a zoom in on the pitcher
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Few things.. notice that he's right at the apex of his delivery? Yeah, I'm neither that good nor that lucky. That point in a pitching delivery also has the least amount of motion, so that's what HDR+ picked as the base image. Catching that with exposure bracketing HDR would have been damn near impossible.
Sorry.. started to go full on geek...