Is osmosis really a special case of diffusion across a semipermeable membrane?
I've listed a series of points about diffusion and Osmosis.

The cause of osmosis is simply diffusion: the solvent can diffuse through the membrane and the solute cannot. 

Thermodynamically, we say that the reason that the diffusion occurs is that the chemical potential of the solution is at a lower chemical potential than the pure solvent and so there is a tendency for the solvent to pass into its solution and to dilute it. The osmotic pressure is the excess pressure just needed to stop that flow. If more pressure were to be applied the flow of solvent would be reversed and the solution would become more concentrated.  

Osmosis occurs because there is an increase in entropy overall when the solvent dilutes the solution. There is a decrease in entropy in the pure solvent side because its volume decreases but this is balanced by an increase in entropy on the solution side as the volume increases so this is not the reason osmosis occurs. The reason is the increase in entropy that is experienced by the solute now in a larger volume or equivalently more solvent molecules in which to dissolve. This means that there are more ways of arranging solute and solvent molecules compared to that before osmosis stared, and this represents an increase in entropy. (Left unchecked this will continue until the solution is so dilute it is indistinguishable from the solvent).

When we say that a solution has an osmotic pressure of, say, 5 atm, it does not mean that the solution exerts this pressure but that if it is in contact with its solvent through a semi-permeable membrane then an applied pressure of 5 atm is needed to maintain equilibrium.

It is possible for a solute in a solution to diffuse spontaneously from regions of higher to lower concentration so that uniform concentration is reached and the solution is at equilibrium.  Naturally, diffusion does not stop at this point but is always present both for solute and solvent. 

In some cases, however, diffusion can occur form lower to higher concentrations. Such a situation is that when a solute is dissolved in two immiscible solvents in contact with one another, and in which the solute has different solubilities. Solute diffusion will occur until the chemical potential in each solvent is equal which can mean diffusion into the more concentrated solution. Thus at equilibrium there will be a large difference in solute concentration but no overall flow of solute due to diffusion.