It's true that decades ago polyester thread was so strong it was capable of cutting through cotton. However, today's world of thread is completely different and that no longer holds true. This Youtube video demonstrates how the tensile strength of today's polyester is very similar to cotton:
https://www.youtube.com/watch?v=9W5OxygKlUk
I like Aurifil 50wt 2-ply cotton thread for piecing (Prescencia also gets very high marks for piecing). These are not the typical 50wt 3-ply threads most of us are used to using; they are much thinner and much smoother, producing considerably less lint. When piecing with cotton thread, a huge advantage to using these threads is that they take up less bulk in the seam, making it easier to achieve perfect 1/4" seams that iron flatter. I like cotton thread for piecing because it has a little "grip" so that, when chain piecing, the ends are less likely to un-sew after cutting. For me, they also result in a flatter seam with less puckering than polyester threads of similar weight, at least with my favored sewing machine. However, these days, it is really a matter of personal preference what kind of thread is used for piecing, especially since there are variations in machines as well as people.
For quilting, I have come to much prefer polyester threads. They produce less lint than even the highest quality cotton threads. An added benefit of polyester is that it simply lasts longer than cotton. Jamie Wallen mentioned that he has visited a lot of museums and talked to a lot of curators who bemoan the fact that they have many interesting quilts in storage that they are unable to place out for display because so much of the cotton thread used to quilt them has dry-rotted. This is why Jamie uses polyester thread for his quilting (unless a customer insists on cotton thread).