CyanogenMod isn't today what it was prior to 2013. Founded in 2009 by Steve Kondik, CyanogenMod is a community developed custom ROM that offers many advantages over stock Android such as performance and security upgrades, along with further options for customization. Like Android, CyanogenMod is open source. Meaning anyone can contribute code to squash bugs, add features etc. Added code can be tested further, commented on and voted up or down. Lead developers over the project have final say as to whether or not the code will be added to source and used in official builds. Between the years of it's founding in 2009 and it's incorporation in 2013, CyanogenMod became increasingly popular amongst the Android Community, and the potential for growth, and profit, was too much to be ignored.
With funding from Venture Capital, Steve Kondik and Kirk McMaster founded Cyanogen Inc which they used to push CyanogenOS. As CyanogenOS popularity grew so did Cyanogen Inc. The start-up began to make it's way into OEM territory, forming relationships with smartphone manufacturers. The first ever device to be shipped with CyanogenMod on board was a special edition Oppo n1. Keep in mind that the device did in fact ship with CyanogenMod, not Cyanogen OS. Coincidentally, or not really, former vice president of Oppo, Pete Lau, founded an Oppo sister company, OnePlus. With close relations already, Cyanogen Inc partnered with OnePlus and offered it's software to be on board OnePlus' first device, the OnePlus One. This time not as CyanogenMod but CyanogenMod 11s, which is now known as Cyanogen OS 11. Cyanogen OS 11 is based off of Android 4.4 KitKat. So the question remains, what's the difference between CyanogenMod and Cyanogen OS, and what does Cyanogen Inc have to do with it?
CyanogenMod was and still is the open source, anyone can build it, distribute it and flash it onto the their device open source project that started in 2009. Developers VOLUNTEER their time and resources maintaining CyanogenMod for free and allow unrestricted use the world over, with downloads estimated to be over 50 million. These figures do not reflect the number of users who choose not to report their usage statistics. Cyanogen OS on the other hand is a modified, commercial version of CyanogenMod and contains closed source proprietary software that can only be found on the few devices running Cyanogen OS, such as the OnePlus One. While the developers of CyanogenMod are volunteers, and indirectly contribute to Cyanogen OS' development, Cyanogen OS is maintained by paid developers. Cyanogen Inc is simply the business in the front. It oversees the development of both CyanogenMod and Cyanogen OS all while maintaining business relationships with manufacturers who are willing to install Cyanogen OS on their devices and purchase licensing to distribute Cyanogen's proprietary software. Current and future business plans for Cyanogen Inc include partnerships with Qualcomm, Microsoft, Boxer and Micromax. Oh and it's ambitious over taking of Google as the primary distributor of Android based devices.
Cyanogen Inc maintains that it is still very much the same community driven and free to use operating system it started out as. Anyone with the knowledge and a supported device can download and install CyanogenMod. But you'll be left without their exclusive software that makes CyanogenMod Cyanogen OS. Much in the same way that Google purchased Android, developed it further and added it's own software for profit, Cyangen Inc took CyanogenMod, developed it further and added it's own software for, you guessed it, profit.