Usui Reiki is of the lineage directly handed down through Master Mikao Usui. He was the founder of Reiki as we know it today. As it has been handed down it has grown and changed a bit, especially since it has come to the West, but fundamentally, it is the Reiki practice that is recognized as from the founder.
Karuna Reiki is the embodiment of compassionate action, and was developed by William Rand and 12 other Reiki Masters involved in the International Center for Reiki Training.
It is based on the idea that energy is not stagnant, and is increasing in vibration as time progresses. In this aspect, Reiki energy grows and increases in vibration. So, Karuna Reiki is a bit more intense energetically, and is only taught to Reiki Masters.
If you are interested, I can explain how it came about, but this is the short answer,
Thank you for your interest and being unafraid to ask.