It won't win "most compact", but it might prove more efficient than many:

#! perl -slw
use strict;

sub z {
    my $soFar = 1;
    my $lastC = substr $_[0], 0, 1;
    my( $lastP, $bestN, $bestP ) = ( 0 ) x 3;

    for my $p ( 1 .. length( $_[0] )-1 ) {
        my $this = substr $_[0], $p, 1;
        my $d = abs( $this - $lastC );
        if(  $d != 1 and $d != 9 ) {
            $soFar = 1;
            $lastP = $p;
        }
        ++$soFar;
        if( $soFar > $bestN ) {
            $bestN = $soFar,
            $bestP = $lastP;
        }
        $lastC = $this;
    }
    return $bestN, $bestP;
}

while( <DATA> ) {
    chomp;
    print;
    my( $l, $p ) = z( $_ );
    print ' ' x $p, substr( $_, $p, $l ), "\t", $l, "\n";
}

__DATA__
01234567890123456789
78901234567890123456
98765432109876543210
21098765432109876543
01234567890987654321
012345678890123456789
789012345677890123456
987654321099876543210
210987654322109876543
012345678900987654321
012345678909876543210
[download]

In the absence of an answer re: 090, I've noted your abs(1) criteria. Also, first of equal longest.

Could be simplified if you don't need to know where the longest occurs.

Output:

C:\test>junk
01234567890123456789
01234567890123456789    20

78901234567890123456
78901234567890123456    20

98765432109876543210
98765432109876543210    20

21098765432109876543
21098765432109876543    20

01234567890987654321
01234567890987654321    20

012345678890123456789
         890123456789   13

789012345677890123456
78901234567     11

987654321099876543210
98765432109     11

210987654322109876543
21098765432     11

012345678900987654321
01234567890     11

012345678909876543210
012345678909876543210   21
[download]