CyanogenMod is known for its customisability, enhanced security and additional features, while MIUI focuses on ease of use in addition to offering enhanced native apps, better notifications and things like call recording and call blocking.
Both operating systems support themes and let users customize all UI elements. Of the two, Xiaomi's MIUI content store has richer themes and wallpapers. It's worth pointing out that there's an ongoing dispute between OnePlus and Cyanogen due to which OnePlus One will not receive any software updates in the future.
However, OnePlus is working on its own Android ROM (based on Android Lollipop) which is expected to release in February. The new software may be devoid of some CyanogenMod features.