With Name::Name->subroutine() you need to capture the class name as the first argument in subroutine() method.

package Name::Name;
sub subroutine {
  my $class = shift; # shifting @_
  print $class, "\n";
}

package main;
use Name::Name;
Name::Name->subroutine();

# output:
# Name::Name
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It's callled class method calling. With Name::Name::subroutine(), you call subroutine() with fully qualified package name without any argument.

I'm not sure what's missing, but if the subroutine in question is new() then you can't call it Name::Name::new() as it expects the class name as its first argument. Similiar way to the class method calling above is Name::Name::new('Name::Name').