There are 2 places where you can check a kick : the knee and the shin. If you check with your own shin bone, you are creating a shin to shin contact and, intuitively, one can expect the damage to be similar for both opponents. However, while the location of the hit will be similar, the results, at least if you want to talk about physics, will be very different. So yes, the same amount of force it applies, but where it goes and what it does can and will vary a lot.
On the kicker's part, you have a lot of momentum going through your leg. Once the hit occurs this momentum translates into a large amount of stress in your shinbone, in the form of a bending moment. If there was enough momentum, this can cause the shinbone to snap, as bones are mostly made to resist compression rather than bending.
On the checker's part, we need to remember that the knee is bent, but not fully. When the kick is received, the impact will cause the leg to bend further, up until the calf potentially collides with the thigh. This bending of the leg acts a bit like a cushion, absorbing part of the hit's energy. As a result, the bending moment on the defender's shin is greatly reduced and thus will rarely result in a snapped tibia. Also, the checker usually checks with the upper shin. Mechanically speaking, this also causes a reduction of the bending moment, resulting in lower stress for the bone.
The other possibility is a check with the knee. In this situation, the kicker's outcome is mostly the same, but the checker's is very different. As was previously mentioned, bones are designed to withstand tremendous compression forces. When checking with the knee, the force is mostly transferred to the femur, which is probably one of the strongest bones in your body (it is also the largest). If done properly (the check done in the UFC fight you mention being a very good example of proper technique), the force can all be absorbed by the femur with a greatly reduced risk of injury on the defender's part.
Why do this instead of receiving the kick on your thigh? The kneecap and shinbone used to check the kick are pretty much sticking out of your body, meaning you will receive the kick straight on the bone, and not on muscle tissue. A strong kick on the back or side of the thigh can cause lots of damage, which will result in a reduced ability to fight (less powerful kicks, reduced balance, impaired mobility, etc.). Checking with a bone reduces the pain (less pain receptors in a bone than in a muscle) and the long term damage that you will endure in the fight, provided that your leg doesn't snap. But as was mentioned, it is more likely for the kicker's leg to break before the defender's does.
TL;DR - Checking the kick redirects the blow to an area that can more easily withstand the hit or absorb part of the damage, at least compared to the kicker. When executed properly, it will likely hurt the kicker more than the defender.
