adding to your list of test cases:

foreach my $n ( @ARGV ? @ARGV : (20, 48, 96, 7, 69, 33, 13195,  600851
+475143, 2147483647) ) {
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... and running with some different test cases, I get

perl pm1194192.pl 18 54 27000 29474100
Number: 18 , Prime_Factors: => 2 3 3
Number: 54 , Prime_Factors: => 2 3 9
Number: 27000 , Prime_Factors: => 2 2 2 3 5 9 25
Number: 29474100 , Prime_Factors: => 2 2 3 5 15 32749
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That's not right for all cases -- any that happen to have more than one instance of a non-two prime. 18 sneaked by, because of the way the error works, but it becomes obviously wrong for 54: 9 is not prime, and should have been eliminated during the $i=3 iteration. In 27000, 9 and 25 are both non-prime. The redo in the example code in Cristoforo's post handled multiple instances of the same prime factor, but your for loop does not

If I change the next to redo, it fixes that problem. And if I take the root after every recalculation of $num, it becomes somewhat more efficient:

$ time perl pm1194192.pl 18 54 27000 29474100 3217491003
Number: 18 , Prime_Factors: => 2 3 3
Number: 54 , Prime_Factors: => 2 3 3 3
Number: 27000 , Prime_Factors: => 2 2 2 3 3 3 5 5 5
Number: 29474100 , Prime_Factors: => 2 2 3 3 5 5 32749
Number: 3217491003 , Prime_Factors: => 3 32749 32749

real    0m0.010s
user    0m0.008s
sys    0m0.000s

$ time perl pm1194192.pl --sqrt 18 54 27000 29474100 3217491003
Number: 18 , Prime_Factors: => 2 3 3
Number: 54 , Prime_Factors: => 2 3 3 3
Number: 27000 , Prime_Factors: => 2 2 2 3 3 3 5 5 5
Number: 29474100 , Prime_Factors: => 2 2 3 3 5 5 32749
Number: 3217491003 , Prime_Factors: => 3 32749 32749

real    0m0.008s
user    0m0.004s
sys    0m0.000s
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When I'd run without --sqrt, I'd consistently get real times of 10-15ms and user times of 8-12ms. With --sqrt, I consistently got real 7-9 and user of 4ms.

For fixing the multiple of the same prime factor problem, instead of using redo, you could also change if ($num % $i == 0) { to while ($num % $i == 0) { (and my sqrt mod would still work as advertised)

---

update: or, with proper Benchmarking of two different copies of the function (prime_factors() without the per-loop sqrt, and prime_factors_sqrt() with per-loop sqrt):

...
my @list = (20, 48, 96, 7, 69, 33, 13195,  600851475143, 2147483647, 1
+8, 54, 27000, 29474100, 589806);
foreach my $n ( @list ) {
    my @result = prime_factors($n);
    print "Number: $n , Prime_Factors: => @result", "\n";
}

use Benchmark qw(cmpthese);
cmpthese( -10, {
    once =>  sub { prime_factors($_)        for @list },
    every => sub { prime_factors_sqrt($_)   for @list },
});
__END__
...
        Rate  once every
once  16.9/s    --  -94%
every  286/s 1593%    --
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