I know that steel is generally iron with carbon and probably some other things in it, but I'm doing a research paper on architecture in Tokyo, and touching on the Tokyo Tower.The Tokyo work is larger than the Eiffel Tower by 12 meters, but lighter by 3000 tons. The only reason I can find for the difference in weight is that the Tokyo Tower is made of steel and the Eiffel Tower in made of iron, and since the Tokyo Tower is modeled after the Eiffel, there's really not a substantial difference in the amount of metal used....
For the same volume of part, like a cubic inch, the weight is very close. If you base the comparison on strength, like a bar strong enough to hold XXX pounds, then the forged bar will be lighter, smaller still hold the weight.
Steel and iron weight about the same, or more correctly, have about the same density. Steel is stronger, which allows structures to be built with less material, and thus lighter.
There are lightweight steel alloys that are lighter, for a given volume, than pig iron. The words iron and steel have referred to different materials at different times, and are used in different ways. A steel structure may also be lighter because steel alloys are stronger, in a given cross-section, that iron or other types of steel. For example, a 1-inch bar of chrome-vanadium steel is stronger than a 1-inch bar of 4130 mild steel or pig iron. The more you find out about it, the more complicated it is. But the answer to your question is (a) Yes, and (b) it's all relative.
