Let me be more explicit. If you change something from a simple accessor that you reach through the tie methods like $foo{'bar'} = 7 to a method call, you will have to either change that calling code to something like tied($foo)->set_bar(7) or else put a special case in the FETCH method so that it calls "set_bar()" if the key is "bar". This would break the abstraction in the former case or compicate the code in the latter.

UPDATE: as requested, an example, using your code. Say we change the definitions of get/set_foo in MyThing to something like this:

sub set_foo {
  my ($self, $newfoo) = @_;
  $self->[1] = localtime(); # track last change
  $self->[0] = $newfoo;
  $self->_recalculate_bar();
}

sub get_foo {
  $self->[2] = localtime(); # track last access
  return $_[0]->[0];
}
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After this change, the $clunker code using MyThing requires no changes at all. On the other hand, the tied HashAPI version, must change the accessing code from this:

$slick->{foo} = 'Some value';
print $slick->{foo}, "\n";
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to this:

$slick->set_foo('Some value');
print $slick->get_foo(), "\n";
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You could avoid needing this change by changing the FETCH and STORE, like this:

sub STORE {
  my ($self, $key, $val) = @_;
  # assuming this class has more than just 'foo'...
  unless (grep { $_ eq $key } @CLASS_KEYS ) {
    croak (ref $self) . "has no $key member";
  }

  if ($key eq 'foo') {
    $self->[1] = localtime(); # track last change
    $self->[0] = $val;
    $self->_recalculate_bar();
  } else {
    # set normal keys somehow
  }

}

sub FETCH {
  my ($self, $key, $val) = @_;
  # assuming this class has more than just 'foo'...
  unless (grep { $_ eq $key } @CLASS_KEYS ) {
    croak (ref $self) . "has no $key member";
  }

  if ($key eq 'foo') {
    $self->[2] = localtime(); # track last access
    return $self->[0];
  } else {
    # get normal keys somehow
  }
}
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That would quickly become cumbersome though.