# First suggestion: add error checking.
open (ONE, $file1) or die "Couldn't open $file1: $!\n";
open (TWO, $file2) or die "Couldn't open $file2: $!\n";

# 2nd suggestion: check syntax.
while ($line1 = <ONE>) { 

    # 3rd suggestion: eliminate unneeded loops.
    $line2 = <TWO>; 

    # 4th suggestion: don't chomp if you don't need to.

    # 5th suggestion: If you don't need the whole array, don't keep it.
    $element1 = (split ' ', $line1, 2)[0];  
    $element2 = (split ' ', $line2, 2)[0];

    # 6th suggestion: Handle either or both cases. 
    if ($element1 ne $element2) {  
        print "$element1 does not equal $element2 at line $.\n"; 
    } else {
        print "$element1 is the same in both files at line $.\n";
    }
}