Slight note first - I avoid baseball, but I was informed that neatsfoot oil is also used on baseball gloves.
Sweet iron used to mean low carbon wrought iron, from the Spanish "hierro dulce." These days it's used to refer to cold-rolled mild steel or low carbon steel.
It's used in bit mouthpieces, mostly by western types, because a lot of people think the slight rust that forms encourages acceptance of the bit. (That's not why it's called sweet, but is why the term is generally used over "mild steel"). In other words, like copper, it appears to taste better and be more accepted by some horses over stainless steel.