Stirring may not mix the ingredients depending on condition. Or, it may just be moving the liquid in a continuous circle with no real change in the up or down direction. Mixing implies an intermingling of ingredients whereas stirring does not. Both use just a rotor. Homogenizing blending imply shearing forces as well that can change any particle size present if set up that way, or it may imply simply dispersing the ingredients forcefully.

The three are not necessarily identical, but many of the effects involved are the same. Mixing does not imply any shearing or blending force, nor does it involve heat. A blender may generate quite a lot of heat and may boil the liquid(s) involved. So, there is also a difference in speed.