The suffixes -kinesis and -taxis both refer to movement. Kinesis is actually more random than taxis.
Example: Cytokinesis is the splitting of a cell into two new cells the cells are moving away from each other but to no particular place.
Phototaxis refers to an organism moving towards light as a stimulus. The motion is specific to the light stimulus and it is not a random movement.
Chemotaxis is movment toward or away from a specific chemical.
Photokinesis is any non specific movement in response to light.
The site included provides several uses of the suffix -taxis.
Kinesis and Taxis are both terms that relate to how an organism responds to positive or negative stimulation. The difference between the two is that with kinesis the organism responds by moving either toward or away from the source of the stimulus. Taxis has the organism just moving randomly.
This concept is easier to understand when looking at examples:
1) A big ugly bug is sitting there and you shine a flashlight at it. The bug, because of some instinct, walks toward the light (or flies, like a moth.) Its direction is based on where the light is coming from. This is kinesis.
2) Different ugly bug doesn't like the smell of wet dog. The bug's instinct will make it run around randomly trying to find a spot where it doesn't have to smell wet dog. This is taxis.
Do you see the difference? One has movement directly toward or away from a stimulus, while the other one has movement that is caused by the stimulus but the direction of which is random.
Hope this helps!