destroyis correct method for removing an entity directly (via object or model).
Example:
$teetime = Teetime::where('date', '=', $formattedDate)->firstOrFail();
$teetime->destroy();
deletecan only be called in query builder
Example:
$teetime = Teetime::where('date', '=', $formattedDate)->delete();
From documentation:
Deleting An Existing Model By Key
User::destroy(1);
User::destroy(array(1, 2, 3));
User::destroy(1, 2, 3);
Of course, you may also run a delete query on a set of models:
$affectedRows = User::where('votes', '>', 100)->delete();
More info: http://laravel.com/docs/eloquent