The core_ webservices are mostly exactly the same as the moodle_ webservices, there are slight variations with at least one, which one(s) I cannot remember at the moment.
Eventually the moodle_ webservices will be dropped, in name only, and the core_ webservices will continue.
The change happened between Moodle 2.0 and Moodle 2.1. If you are using Moodle 2.0 then the core_ webservices will not be present and you will need to upgrade.
Any new software which uses the Moodle webservices should use the core_ versions in preference.