"Stronger security" with Drupal. What, you mean Joomla can be more easily hacked?
Well, no, that's not typically what the Drupal people mean by this. As with all CMSs, the number one weak point for hacking is passwords that are easily cracked. Since that's all about human error, Joomla and Drupal are equally vulnerable in that regard.
The second weak point in the chain is making sure that your web developer updates your site quickly and efficiently to the latest "dot release" as they're released, rather than ignoring sites as our competitors frequently do. Open source CMSs are great in that they're free, and freely available, but it also means when a security vulnerability is found, everyone knows about it. When the latest security release comes out, you need to upgrade your CMS, no matter what package you're running, or you're more vulnerable to being hacked. We at 4Web include free upgrades for a year as part of our contract, and we stay on top of when new releases come out and get the upgrades done -- unlike many of our competitors. You do this on your Windows/Mac computer, often unseen, with little updates all the time (largely security-related). Your CMS software is really no different.