Lyocell is a cellulose fiber (more on that later) derived from sustainable that are harvested and certified by Lenzing's standards. Here is a link to Lenzing as you can read more about them, Tencel and Modal! It's a fully closed loop cycle, meaning resources used go back into the process and reusing up to 99.9% of water. The yarn is strong, fabric is soft and biodegradable like hemp and other natural fibers. It is also naturally resistant to bacteria and thermal regulating.
Tencel lyocell is softer than silk making it very gentle on the skin. The process allows the dyed fabric to be vibrant in colour while also providing many different colourways.
Picture courtesy of Lenzing.
So, Lyocell is the actual fiber while Tencel is a brand of Lenzing. To use the "Tencel" name Lenzing, licencing is needed from them.