My solution to the problem isn't as compact as the other solutions, but TIMTOWTDI!

#! /usr/local/bin/perl

$string1 = "the date is today";
$string2 = "the date is tomorrow";

$concat = $string1 . "#" . $string2;

if ( $concat =~ m/^(.*)(.*)#\1(.*)$/ ) {
    print "matching part: $1\n";
    print "difference: $2\ndifference: $3\n";
} else {
    print "strings $string1 and $string2 do not match!\n";
}
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When run it generates the following output

matching part: the date is to
difference: day
difference: morrow
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I have to note that this solutions uses the fact dat the to strings doesn't contain any poundsymbols (#). When one of the to strings contain a poundsymbol, you have to use another symbol(sequence) to, for example:

...
$concat = $string1 . "#!!#" . $string2;

if ( $concat =~ m/^(.*)(.*)#!!#\1(.*)$/ ) {
...
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I hope that this will give you a hint to construct your own solution!