There is steel jacketed ammunition, with lead core, like most U.S. GI 30-'06 and 7.62 NATO, used to save copper.
There is ammo with both steel core and steel jacket, like much German WWII ammo, used because of the scarcity of lead and copper.
There is armor piercing ammo, which has a hardened steel core.
So, AP ammo has a steel core, but not all steel core ammo is AP.
AP penetrates not only because of the hardness of the core, but because the bullet stops on the armor plate, and its energy is converted to heat, which melts a hole in the plate, allowing the core to penetrate the partly molten steel. An ordinary bullet will cause the melting, but there will be nothing left for penetration.
Jim