The simple answer is that it was too heavy.

Weight was one of the main reasons that armour fell out of use in the first place. It's not that it was ineffective - even thin steel will provide some protection - but the cost-benefit ratio changed, compounded by changes in warfare, such as campaigns over greater distances requiring greater mobility. There's an excellent explanation of this by /u/WARitter here and a shorter and not-quite-so good answer from me here. To protect against musket fire, armours got up to 4mm+ in thickness, which is about the limit of what is wearable for an infantry soldier.

Essentially that cost-benefit analysis remained much the same for infantry until the post-war period, when new materials like kevlar and ceramics came into use. The resistance of steel increased somewhat, but the penetrative power of weapons also increased. Look at this as an example of a contemporary steel armour plate that is proof against rifle fire. At 0.26"/6.6mm thickness and 3.6kg/8lbs, it's comparable in weight to armour from earlier periods.

There are some examples of body armour being used in particular situations in WW1&2. Some cavalry and even some colonial infantry units appear to have worn cuirasses at the start of WW1, German snipers were apparently sometimes issued body armour, and Russian assault troops sometimes used it in WW2. These are isolated example though.