API 5 corresponds to minecraft 1.10 not 1.8.9
When you download SpongeForge / SpongeVanilla you can see the API version used.
Image taken from the SpongeVanilla download site.
API 6 & API 5.1.0 are not yet released/stabilized.
Since API 6 is a Major Version change that means that changes made in 6 will be incompatible with some plugins.
Since API 5.1 is a Minor Version change, theoretically plugins made for 5.0.0 are guaranteed to work on 5.1.0
Plugins made for 5.1.0 may not work on 5.0.0 depending on if they use the new features.
Before an API version is released, changes can happen unexpectedly, something that works on 6 or 5.1 today, might not work on 6 or 5.1 tomorrow.
The first listed (and most recently updated project worlds) is made for API version 5.
The second listed (and somewhat dated) is made for API snapshot 4.2.0 which (from memory) I don't think saw release, but many people were using anyway.
I hope this not only clears up what downloads to use for which version, but gives you some insight into the versioning systems used.
We are currently working on making this a lot easier to understand.