The difference between iron and steel is that in steel, the carbon is dissolved into the iron matrix. As mixdenny notes above in pig-iron the carbon exists as separate nodules that are not integrated into the iron matrix.
Ignoring the complexities of carbidies…
The dissolved carbon exists as individual atoms in the steel. This brings about the single most important characteristic of steels. Where in the matrix of iron atoms do the carbon atoms sit? An iron metal matrix is a cubic lattice. Think lots and lot and lots of cubes, all the same size, all neatly stacked in 3D. The iron atoms can be though of as sitting at the corners of the cubes. So how do you fit the carbon in? There are two things to notice. You can’t fit a carbon atom in without distorting the matrix. Those iron atoms will pack as tight as they can, so the gaps are small. The next thing is that there are two places you can stick a carbon (or other alloying element) in the matrix. You can stick it in the centre of the cube, so that it is surrounded by the 8 iron atoms at the cube’s corners. Or you can stick it in the middle of one of the faces of the cube, so that it is surrounded by the four atoms that make up a face. Unsurprisingly these two arraignments are called body centric and face centric respectively.
You can’t get much carbon into the steel, the distortions of lattice reach some way. What matters is that this distortion of the lattice acts to stress it. And because of this, the overall properties of the lattice change. The body centric configuration is very highly stressed in comparison, and becomes much harder.
The really neat trick is that it isn’t hard to control where the carbon atoms are sitting. If you heat the steel up to a high temperature the carbon atoms become quite mobile and will rattle about all over the place. If you slowly cool it down, they will tend to settle in the minimum energy configuration, which is in the centre of the faces. So the steel is dominated by face centric carbon. However if you cool the steel down very quickly you can catch a lot of the carbon atoms in the body centric location, and get the steel cold enough that the carbon atoms remain trapped there. So you can make the steel have a range of mixtures of face centric and body centric configurations. These two configurations have different properties. Managing the balance of the two is a critical part of hardening and tempering steel.
It turns out that the two configurations are evidenced by macroscopic properties of the lattice, and inspection of the metal with a microscope will reveal different appearances of the crystal structure. This is how the two were first discovered. The usual name given to the two structures are named after their discoverers, Adolf Martens and Sir William Chandler Roberts-Austen. So we have Austenite (face centric) and Martensite (body centric). These terms have no become applied to pretty much any other metal alloy for which similar configurations are found. However the relative strength and hardness of the Austentic and and Martensetic states varies from metal to metal.
To get hard steel you need to get the carbon content up enough that the above manipulations make any difference. Hardening steels have more carbon. Or you can diffuse carbon into the structure. Cooking steel with carbon (and a reducing atmosphere) can allow carbon to slowly diffuse into the surface, yielding an object that can have its surface hardened. But it takes hours, has to be red hot, and you need the reducing atmosphere. So, no matter what you do with seasoning your pan, you won’t be doing much to the actual iron.