Diffusion is, on the whole, an effective way to transport essential nutrients into a cell, and take unwanted wasteout.
For unicellular organisms like amoebas and algae, this satisfies all of the cellular requirements, but for larger, more complex cells, problems can arise.
The characteristic of a cell which determines how effective diffusion is, is the surface-to-volume ratio. Essentially, a cell with a large surface area will be able to more quickly and efficiently absorb molecules through the membrane than a cell with a small surface area, but both cells may have the same sized volume.
Not the easiest thing to explain. Check out this picture to make it a bit clearer:
Another limitation is that diffusion as transportation is a rapid process which can only occur over shorter distances, and only through thin-walled membranes.