How is CyanogenMod's built-in root access different? I don't know a lot about Cyanogen but realizing Snapchat fails to detect the root in it catches my attention. Does it have a different binary than su for root access? What mechanisms are different which grants an app root access?
Cyanogen has a different root method since version 11 upwards... It is now integrated with the stock privacy manager inside privacy guard !

I believe it doesn't recognize root for the reason that it can be disabled inside development options. 

You need to manually configure it via development options, This means that snap chat wont be able to run a su or adb su / sudo command