What's the difference?

Well, the Aim-At constraint will effect only the single bone that you constrain (well, that and any of its children).  It will never effect, for example, the parent of the constrained bone.

A Kinematic constraint, on the other hand, accesses the Hash programs "Inverse Kinematics" engine.

Inverse Kinematics is the process by which you can tell the computer "I want DojoBot's hand to be right here, and I don't really care how you have to bend the arm to get it there."  Using convoluted mathematics, the computer can solve for most positions, producing an appropriate orientation of the arm that will place the hand exactly where you requested.

Obviously, the Inverse Kinematics engine can effect parents in response to the actions of their children.  For instance, in moving the hand, you effect the position of the forearm and bicep.

Specifically, Inverse Kinematics (or IK) will change the position of a parent bone if its child bone is considered "Connected" (by clicking on the Connect button in Bones mode).  And then, if that parent bone also has a parent to which it is connected, the IK engine will repeat all the way up the hierarchy, until there is a disconnect.

The set of bones that are influenced in this way is typically referred to as a "Kinematic chain."

So, to get back to the subject at hand, the Bowing and Breathing tutorials use Aim-At constraints because each bone of the spine is individually controlled in those tutorials.  There is no need to activate the IK engine.

In this tutorial, however, we are constraining only the tips of the fingers.  The rest of the motion we count on the IK engine to supply.  By using the Kinematic Constraint, we explicitly tell the IK engine to solve for a position which places the end of the tip of the finger at the base of its specified target.  It is the mathematics of that IK engine which produce the gentle curves that the fingers take on in our demonstration.
 
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