Update: Added minor optimisation. Update2: Rolled the optimisation into the while loop.

Something like this?:

#! perl -slw
use strict;

my $seg1 = "The man who likes reading books and writing poems.";
my $seg2 = "The man who likes reading big books and poems.";

my $best = '';
while( length( $seg1 ) > length( $best ) ) {
    while( $seg1 =~ m[(?!\s)(?=(\b.+\b)(?!\s))]g ) {
        my $bit = $1;
        $best = $bit if $seg2 =~ m[\Q$bit] and length( $bit ) > length
+( $best );
    }
    $seg1 =~ s[(?:\s|^)\S+$][];
}
print $best;

__END__
[17:04:21.46] C:\test>junk39
The man who likes reading
[download]