Why are people with full suspension bikes with several inches of travel be concerned about 'Stiffness' when discussing the advantages of thru axles. I would think this is more of a concern for rigid bikes which don't have suspension. Isn't the advantage that you're just less likely to snap the axle when riding down a gnarly trail not really how 'stiff' the axle is or does this have some other purpose?
Thru axles not only keep the wheel in place, they also strongly join the right and left dropouts together. This structurally reinforces the component either being it a fork or a rigid/suspended rear triangle.
This simply makes the bike able to handle greater forces before failure, because it distributes forces between both sides of the structure. It also avoids twisting of the parts. For example, if one arm of a fork twists, it means it may loose alignment relative to the other, thus meaning the wheel is not pointing in the direction the handlebar tells the rider, this leads to a bad feedback and imprecise ride feel. A thru axle keeps both sides of the fork aligned, eliminating (or reducing) this issue. It also keeps the lower ends (bottles) better aligned with the upper parts (stanchions), allowing for a smoother operation and longer life of seals and bushings.
For suspended rear triangles it also makes the structure stiffer, this allows better distribution of forces through the stays, so those forces reach the shock absorber with better alignment, also allowing for smoother operation and longer component life. It also keeps the pivots aligned, thus giving them a longer life for both bearing type or bushing type pivots.
The bottom line is that a stiffer structure is more efficient and gives better feedback to the rider, allowing for a more confident and precise ride. It also allows a lighter bike to handle greater forces.
Stiffness in this context is the ability to resist twisting and bending forces caused by turning and heavy braking (usually on difficult ground). A stiff shock moves up and down freely but allows no twisting or bending back and forward/side to side.
MTB riding benefits from stiffer forks. The main advantage is the precision of tracking by the front wheel. As an over simplification, think what happens when the forks twist - the front wheel no longer points in the same direction as the handlebars. On difficult ground this can change in a split second, making riding more difficult. The stiffer the fork, the more precisely the front wheel can be placed. In the extreme, a flexible fork can throw a rider offline, as best, slowing him, at worst, causing a prang.
For decades the 9mm QR was the standard for virtually all wheels. As MTB forks became longer travel and lighter, and riding more extreme, it was decided that a stiffer connection between the hub and fork was required. The 20mm through axle was developed. This was rejected by many as too heavy, so the 15mm Through was developed as a lighter weight alternate providing nearly as much stiffness as a 20mm, and virtually no more weight than a 9mmQR.
There are few benefits for a through axle for all but the most extreme or elite riders. The additional strength is a benefit to the most extreme downhill riders. The additional stiffness is the only other real benefit - therefore, any discussion on through axles is incomplete without mentioning it.
There is some controversy around the need for the smaller through axles on all but the very best bikes. Many believe the offer little advantage and many disadvantages over QR for a majority of riders, and are more a marketing gimmick than a real advance in bicycle performance (for average riders).
The stiffness in the front added by the 15mm or 20mm thru axle is very important because it lets the bike go through very rough sections (roots, rock gardens, corrosed terain) without transmitting unecessary feedback on the handlebars. This makes fast riding on rough terrain easier and safer.
In addition, a stiff axle will not allow the wheel to bend laterally too much, even on a smooth terrain, when going fast on a bermed corner etc. This gives more presision and less lost energy.
In any case, snapping an axle, even a 9mm, should be pretty hard. That is unless something has not been set up correctly.