My ideas:

• If you invoke File::Find::find() with the no_chdir=>1 option, $_ and $File::Find::name have the same value. Use local $_=$_; in the "wanted" callback and you can modify it as you like.
• Just remove the unwanted prefix instead of matching the entire name (unless you want to untaint), and use different RE quotes to avoid the leaning toothpick syndrom: s|^C:/users/User/Music/||i;
• Save the path in a variable, pass that to File::Find::find(), and use the variable in the substitution using quotemeta: s|^\Q$path\E||i;
• Your code lacks a check for a successful open, and it uses the "ancient" two-argument open: open OUT,'>','filename.txt' or die "Can't open filename.txt: $!";
• If you would redirect STDOUT when invoking the script, you don't have to work with OUT and open() at all. That would also allow to pipe the output into another program.
• If you want to list only files, test for files inside the callback, before modifying $_: return unless -f $_;

Alexander