You're only matching the regex once, not collecting all instances of the match. The difference gets more pronounced the longer the string becomes:

use strict;
use warnings;
use Benchmark qw(:all);

my $string="He0Hello~~He2W~orld~~He0Hello~~He2W~orld~~He0Hello~~He2W~o
+rld~~He0Hello~~He2W~orld~~";
my $f;

sub invertedCharclass { while($string=~m/He\d([^~]+)~~/g){$f=$1} }

sub nonGreedy { while($string=~m/He\d(.+?)~~/g){$f=$1} }

cmpthese (-10,
    {
        '[^~]+' => \&invertedCharclass,
        '.+?' => \&nonGreedy,
    }
);
[download]

gives this on my machine:

          Rate   .+? [^~]+
.+?   105088/s    --  -28%
[^~]+ 146676/s   40%    --
[download]