Use Set::Scalar.
use Set::Scalar;
my $bigset = Set::Scalar->new(@bigarray);
my $smallset = Set::Scalar->new(@smallarray);
my $difference = $bigset - $smallset;
This assumes though that you don't care about array order, and don't care about duplicates - you're treating them as sets.
Preserving order, and assuming you're on Perl 5.10+, you can just use smart match instead...
my $smallarray_ref = \@smallarray;
my @difference = grep { !($_ ~~ $smallarray_ref) } @bigarray;
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