If you do not care about balanced parantheses it is simple:

use strict;
use warnings;
my $tlp_p2 = "(((((((NOT(tlp_p0(0) XOR tlp_add_p0(0)) AND
                                         NOT(tlp_p0(1) XOR tlp_add_p0(
+1))) AND  NOT(tlp_p0(2) XOR tlp_add_p0(2)))";
my @var = split /\s*AND\s*/, $tlp_p2;
$,="\n";
print @var;
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BUT: already your proposed split for $var1 and $var2 does not have balanced parantheses. In fact, the closing ) at the end of $var2 corresponds to the opening ( at the beginning of $var1 if I have counted correctly. How do you want to deal with that situation?

Remark: If the terms in your equation are always of this type, you can create them bottom-up easily instead of extracting them from the equation...