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.
The 19th century saw the invention of the cotton gin. This machine quickly and easily separates cotton fibres from their seeds which revolutionised cotton production. Thus, the cotton sheet was born. This was the start of a new industry. It became much easier to produce cotton than flax and that in turn made it a cheaper product. Ultimately cotton became the popular choice for bedding, though the name bed linen stuck.