We don't *YET* have a proper fully structured API per-se.
We have ways to access the data from apps, but it doesn't work the way the OKAPI (or GC API) do...No O-Auth etc (yet).
We have various data structures using JSON that can be used to get most things, but there's a little bit of stuff you need to do to have it Authenticate (plus we need to enable your/these apps in our system to be able to use this)
To be frank, there hasn't been the demand to warrant the substantial time it would take to develop and test a full-fledged API, but if you are developing something we can add you to our special app-developers area and discuss the ways we do things.
As for the Groundspeak API, I believe they are not accepting new applicants at the moment.https://www.geocaching.com/mobile/apidevelopers/
A OKAPI compatable API is on my list: http://wiki.geocaching.com.au/wiki/Geocaching_Australia_Development_List
however as it stands it is low on the priority list.
Getting the new server online is #1... (Mid Aug est.)