I'm just curious as to the difference between the public beta vs the developer beta.
IMO, the biggest difference is/will always be the target audience.
The developer beta release is intended for developers to install on their stable of devices for the express purpose of testing their software/apps for compatibility. Over time, having the developer beta for some was a status symbol - kind of a "look what I have that you don't" kind of thing. That was never Apple's intent, but the back channels being what they are, people clamored to get it, and usually found a way, regardless of the usability factor as their "daily driver". The developer beta software still is intended for developers to test their apps.
Apple has now released a public beta for iOS 9. In part, it is to satisfy the cravings of those techies that want to play with the new release early. Generally, as has been the case with OS X public betas, the public betas tend to be a bit more polished, and often lack features that are included in the developer beta, but that could be more problematic to the overall functioning of the software.
As Keith has said, this time around the build numbers for the public beta and the 3rd developer beta are the same, and, therefore, the software is identical. We shall see what the future brings. I fully expect the developer betas to be released more frequently than the public betas.