#!/usr/bin/perl

open( F1, "file1" ) or die "file1: $!";
open( F2, "file2" ) or die "file2: $!";
while ( $s1 = <F1> ) {
   $s2 = <F2>;
   if ( $s1 =~ /^>/ ) {
      print $s1;
   } else {
      chomp $s1;
      print "$s1$s2";  # (update: added "$" on "s1")
   }
}