Mountain bike axle diameters Out of interest I would like to understand the benefits of having a 15mm or 20mm through axle over a 9mm QR (and a 10mm vs. a 12mm TA on the rear).

What difference does it make to the ride?
The claims are that the through axles offer a stiffer fork, giving more precise handling and performance, over a 9mm QR. 15mm through axles evolved as a lighter alternate to the 20mm ones - which are heavy and overkill for XC and most all mountain riding, with a similar weight to 9mm QR but the stiffer performance. 

At the highest end (both bike and rider) I am certain the performance improvements are real and measurable. As you drop down in cost/ rider ability, somewhere there is a crossover point where it makes no difference - the forks performance (or lack of) smothers any real difference. The cynic in me believes that for all but the most elite on the most expensive bikes, the placebo effect of spending $1000's on new kit is more of a driver for the better performance than any real, scientifically measurable difference. Many reports I have read are similar to  "My shiny new $5K toy with 20mm axles is much stiffer and more predictable than my 10 year old $1K bike with 9mm QR. Those 20mm axles are the best!"

One definite advantage of through axles is that there is a documented and real issue with QR on front forks and disc brakes, where the front wheel is ejected under heavy baking due to the torque generated by the disc. This has been half solved with "Lawyer lugs" - the wheel stays on now, but can come loose. This is only a real issue for those with large disk rotors and those that fail to do the QR up properly. The average XC rider on 160mm discs will never see the problem.

A possible disadvantage of through axles are the removal and re-installation of wheels is harder, often needing tools (you can get QR through axles), making less suitable for field puncture repairs etc. You may need a new bike rack (if you use a fork mount style).