I think the weight of the thread is more important than the content.
As far as piecing goes, you want a thin thread so it doesn't take up too much room and distort the piecing. Most people use 50 or 60 wt., often cotton, but it doesn't have to be. I like Presencia but there are lots of Aurifil and Superior fans. When it comes to the actual quilting, it's a matter of the effect you are going for. A thicker thread like 40 wt is going to show up more than a 70 wt. A shiny rayon will have one look while a matte cotton will have another. Matching the thread color to your project will make the quilting less noticeable. You will see texture more than anything else and it will hide any mistakes! A contrasting thread will make your quilting stitches pop!
The only limiting factor with quilting may be that your machine will like one brand over another. Same brands as above.