Off the top of my head, I can't think of an instance where $a = $b would give a different result than $a = "$b".

This will explode horribly if $b is a reference. Compare the following:

[ateague@dingbat mod]$ perl -MData::Dumper -E '$b = [qw/1 2 3 4 5/]; $
+a = $b; say Dumper $a;'
$VAR1 = [
          '1',
          '2',
          '3',
          '4',
          '5'
        ];

[ateague@dingbat mod]$ perl -MData::Dumper -E '$b = [qw/1 2 3 4 5/]; $
+a = qq|$b|; say Dumper $a;'
$VAR1 = 'ARRAY(0x12609e4)';
[download]

The first one prints the array referenced by the reference. The second one simply prints a string containing the memory address of the array reference.