What is the difference in mosM and mOsm units, if there is any at all?
I think the clinical chemistry notation has not been standardized but certainly it has been inspired by chemists.
The capital M in mosM emphasizes that it is milliosMolar. Chemists use capital M to denote molarity.
On the other hand, mosm indicates milliosmolal. Chemists use small m to symbolize molality.
This notation is not universal.
From this slightly dated article on Osmolality, which is behind pay-wall
Osmolality, Harry F. Weisberg Laboratory Medicine, Volume 12, Issue 2, 1 February 1981, Pages 81–85.

Terminology and Units
In 1934, Gamble, et al.,11 introduced the term ”osmillimole” which has been successively transformed into mosM (milliosmolar), mosm (milliosmolal), the noncommittal mOsm, and the present-day mosmol (milliosmole). Raoult’s law states that the lowering of the vapor pressure (one of the four ”colligative”
properties of dilute solutions) is proportional to the mole-fraction (more easily calculated from molality) of solute present in the mixture.