Viscose, Lyocell and Modal have been made from plant sources typically trees. So all of them have equal chemical base and are known as viscose in Europe and rayon in North America, and scientifically they called as regenerated cellulosic fibers.
The subtle alteration between them comes down to the different manufacturing process and their final structure. The manufacturing process and chemicals of viscose and modal are very similar however modal fibers take slightly different steps after spinning such as stretching to increase their segmental orientation and consequently increasing tensile strength. That means these fibers have the potential to be lighter and finer.
The solvent which is used to extract cellulose from wood in Lyocell producing process is vary. On the other hand, they have more uniform internal structure which leads to improve their ability to absorb water.
The following figures are showing SEM images of the internal structure of these three fibers. The dark areas are illustrating the micro-pores where have filled up with water. It is obvious that Lyocell has higher and more uniform moisture absorbency and as a result more breathability.