I believe 4481 is for this NUC only (or at least units with the Iris Pro 580 GPU).
This message was posted on behalf of Intel Corporation
In regard to your inquiry, first I just wanted to let you know that the 21.20 and 15.40, are called the branches of the version of the driver, the actual driver version is the last 4 digits of it, in this case 4481 and 4501.
The drivers 126.96.36.19981 and 188.8.131.5201 are just different drivers versions, the 4501 is an updated version of the previous versions, that includes the 4481 driver, but they will work with both, the Intel® NUC6i7KYK NUC and the Intel® i7-6700K processor, and that is the because of the graphics controller.
The NUC works with Intel® Core™ i7-6770HQ processor, and the graphics controller of that processor is Intel® Iris™ Pro Graphics 580, and the Intel® i7-6700K processor works with Intel® HD Graphics 530 graphics controller, the latest graphics driver for both controllers is 4501.
Regarding your second question, the 4481 driver regardless of the released date, is the same driver, that date is just where it was posted on our web site, but the driver is the same.
On the following link you will be able to see the specifications for both products:
Also, Windows have its own drivers, so maybe, Windows has a newer version for that graphics controller, and that is why it reports that is newer than the one we have.
Any questions, please let me know.