The twill weave is characterized by an over two, under two pattern. It leaves behind a distinctly diagonal design and is typically wrinkle-resistant due to the thickness. Twill woven sheets are prone to shrink more than those with a sateen weave but are also often times more affordable. 

First of all, because linen’s flax fibers are slightly longer than those of cotton and wrapped a little tighter, linen sheets are stronger which means they will last much longer. If softness is a factor for you, you may want to consider cotton, though. Flax fibers are much rougher than cotton resulting in slightly crisper fabric. But the great thing about linen is that it gets softer after each wash, so essentially it gets better with time.