You use 6 sigma to achieve a higher level of quality. In many situations 4 or 5 sigma makes a big difference. For example, if the airlines were able to deliver people to their destination alive only at a 4 sigma level a DPMO of 31.67, or for every 1 million flights there would be ~32 crashes. Assuming there are around 30k flights/day in the US, it takes only 33 days to have 1 million flights, so we'd see something like a crash a day at a sigma level of 4. 

While I don't know the exact sigma value of safely-landed commercial flights in the US, we can back into it using the 30k/day figure. I believe the last major commercial crash occurred in December 2001, which was roughly 6.5 years ago or about 2373 days ago. At a 30k flights/day*2372 = ~71 million flights. 1/ 71,175,000 = 0.000000014, which is a DPMO of .014. Using the percentile to z-score calculator that gets us a z-score/sigma value of 5.67. So the difference between 4 and 5.67 sigma in this case is many thousands of crashes!

When processes happen a lot and an accident can result in death (like airplane flights) a very high process sigma is necessary--hence 6 sigma.