The delete function is a full delete that will remove the data from the database, however there are some links that it won't remove and will need to be manually done, so for applicants with a history you may find this quite a time consuming process, to track and remove each of the things that need deleting.
The Deletion request plugin doesn't fully remove the data from the database, but does put it beyond use.
It doesn't have the restrictions as above of what it won't remove, other than the options set for the deletion request functions itself (i.e. you can set not to remove someone who has an active placement within X weeks).
So for applicants who have had an active history you will find this a lot easier to delete as well as having batch functionality, which would further speed the process up.