How is steel an improvement over iron?
i'm not sure if you mean cast iron or wrought iron.. I'll assume it is cast iron...and I'll assume carbon steel (rather than stainless steel or steel alloy)
the higher carbon content in cast iron makes it brittle. If you make a sword out of it, it break easily.. contd
" The extra carbon in cast iron makes it more brittle, because carbon wants to clump up into lumpy carbides (an iron-carbon mix) or form flat sheets of graphite (pure carbon), and both disrupt the grain of the iron, making the grain irregular, more brittle, and less strong. By contrast, with less carbon in its makeup, the metal in carbon-steel has a more uniform grain structure. "
Basically, it makes steel stronger.
If you are comparing carbon steel and wrought iron, here is an article that explains it: http://mailleisriveting.weebly.com/metallurgy-i-whatrsquos-the-difference-between-iron-and-steel.html
But basically "Ferrite in its pure form has a Brinell hardness of about 80.  Carbon, by contrast, has a hardness of 700 or more.  Meanwhile, pearlite, the alloying of carbon and ferrite, has a Brinell hardness of 200. This means that steel is much harder that wrought iron, but this hardness comes at the price of ductility, which is measured by only 10% elongation per 2 inches in steel versus 50% elongation per two inches in ferrite.  "