The main difference is in the time of usage:
You would use an Adapter during development of your software, in cases you cannot change the existing interface you introduce an Adapter.
The extension object pattern provides for such changes and enables you to e.g. extend already deployed software.
Also, from the Text you posted:
Adapter supports to adapt an existing interface. The Extension Objects pattern supports
additional interfaces. Extension Objects and Adapter can work together in situations
where an object needs to be adapted to an extension interface.