Can someone outline the advantages Insight's API and front end provide? Any related links (especially for Python!) would be greatly appreciated.
The advantage for me, when I was looking for something, was that insight seemed to be the best that was under active development.
The web frontend was also the better looking at the time.
Not sure what documentation you are looking for, but the API is reasonably well documented: https://github.com/bitpay/insight-api/blob/master/README.md
If you want to dig into the code that's another story. It is hosted on github though, so you should be able get into it.