Both osmosis and simple diffusion are transport mechanisms and unlike facilitated diffusion, both can function without using membrane proteins. However, meanwhile, osmosis refers to the movement of water through the membrane, simple diffusion refers to the movement of molecules. One example of the latter is the exchange of CO2 and oxygen in the cells of the pulmonary veins during breathing.
In conclusion:
Similarities:
1- Both are transport mechanisms that work without proteins
2- Because they are diffusion mechanisms they don't need energy (ATP)
Differences:
1- Osmosis refers to the movement of water through the membrane; simple diffusion refers to the movement of molecules
2- Unlike osmosis that occurs in liquid environments, simple diffusion can occur in gaseous media.