If you care about taking advantage of prompt security updates from your distro provider, you will do this as little as you possibly can. Otherwise you need to pick up that burden yourself, and since most of us aren't security pros, it is a weight that should be assumed with great reluctance. You're no doubt familiar with the dilemma: Are you going to attempt to maintain stability (which isn't at all guaranteed) by backporting patches, or annoying your ops people by forcing version upgrades?
To a certain extent, how bad this gets is a function of how out-of-date your distro is. This is where having packages that are 4 years old can bite you -- you end up rolling your own far more often than you really should. Overuse of tools like virtualenv leads to the same problem.
If you're betting your business on this software and are operating at scale significantly larger than the typical Web service, odds are you're not going to live on the distro-provided packages for very long.
Another good reason to stick with RHEL is that they do a very good job of maintaining compatibility between minor releases. You're pretty much guaranteed to be able to install a third-party package across all minor releases. If it installed on 6.0, odds are it will install and work on 6.4 without issue. Others make no such guarantees.
RHEL has way, way better (and accurate) documentation than any other distro I've used, particularly when it comes to unattended installations and server configuration tuning. Other distributions tend to focus much more on desktop users than massive server installations and their particular needs.
It's a common complaint among people that RHEL is always too far behind the curve in terms of software packages. But really, your OS will be stale the day after you install it, much like a car loses so much of its value the day you drive off the lot. Freshness is a tempting siren, but can lead you onto the rocks.
If you're a sane implementor, you're going to stick with any distribution you choose over the course of several years. You will favor uniformity over freshness, and stability over constant (and arguably needless) work. And no matter what distribution you choose, you are very likely to roll your own packages of whatever software is critical to your business, as you'll need custom patches and so forth.