I'm not sure that's the complete explaination as you get different results if you reverse the order of the statements:

#! perl -sw
use 5.010;

$data = do { local $/ = undef; <DATA> };

my $x = [
        scalar($data =~ s/(\w+)/$1/g),
        scalar($data =~ s/\n/\n/g),
];

say "@$x";

my $y = [
        scalar($data =~ s/\n/\n/g),
        scalar($data =~ s/(\w+)/$1/g),
];

say "@$y";

__DATA__
Line1 Word Something
Line2 Other Word
[download]

Gives:

c:\test>783947.pl
6 2
6 6
[download]

(Plus tr isn't a regex!)