If you're looking specifically at the stable builds of CyanogenMod (CM), then you would be better off rooting Cyanogen OS. Otherwise you would likely be reverting to an older Android version or waiting months for the stable release of the current version of CM to be released. In my experience, the stable release typically comes at the end of the development cycle for a particular Android version, e.g. around when the new version is being released. On the other hand, for the OnePlus One the comparatively stripped out CyanogenOS build would come a few weeks after the release of a new version of Android.
The obvious pro for CM is that it is much more recent and updated. New features are being added constantly, so by the time it reaches a stable release it is much more fully featured than the comparable Cyanogen OS release. The obvious downside is the time that it takes to get to that point.
The obvious pro for Cyanogen OS is that it is built specifically for your device under contract by the manufacturers. It's already on your phone and is supported officially. Being an OEM ROM, it often comes with additional Cyanogen OS-specific apps and themes that aren't available to CM, CM-based ROMs, and AOSP-based ROMs. Although from my experience with the OnePlus One, this point is insignificant.
Between those two options, there isn't likely to be any significant difference in stability. With CyanogenOS, you WILL have to reroot the device every time you flash an official update. You would not have to deal with the steps of the root process that revolve around drivers, which is most of them, but it will also overwrite your recovery so you would have to flash that again as well. With CM the update process involves rebooting to recovery and installing the .zip file. The more I think about this, the more I'd recommend CM. You've changed my mind.
If you are interested in some of the additional features that a custom ROM has to offer, you should browse the lettuce xda page here for custom ROMs. The only people that will be able to answer your stability questions are other people on your same device running these ROMs. A CM 13 build was just released last week so it may be too early to consider that a daily driver but the CM 12.1 build has been around for a while. There are also plenty of other ROMs out there if you want to go down that route.
The rooting process for your device is the same as many others and is very straightforward. Here is a detailed guide. These steps will be common to both rooting CyanogenOS AND installing CM, which is only four additional steps: download ROM zip, reboot to recovery, flash the zip, and wipe cache and dalvik. Reboot.