I recently picked up the ability to upgrade equipment from the Railroad, and got my hands on some clothes and under-armor to see exactly what the stats are on these items when upgraded. All of the items I looked at were exactly the same: at Rank 4 of Armorer, you can give any upgradeable clothing and under-armor 110 Ballistic and 110 Energy Resistance. On clothes, I don't imagine that this is a huge bonus, since you can only wear the one item on it's own; I would imagine that a full set of fully upgraded Heavy-tier armor would provide similar or better stats.

The interesting thing, though, is that Under-armor items get the exact same stats, but you still have the ability to equip armor items on top of them. In comparison, the Vault Suit tops out at 25 Energy Resistance and 25 Radiation Resistance. While the Vault Suit has Radiation Resist that no other under-armor has, the fact that everything else gets far better Energy and Ballistic Resistance makes upgradeable under-armors the better choice once you get the ability to upgrade clothing from the Railroad.

Now, by way of comparison, a full set of Heavy Combat Armor without any upgrades gives 101 Ballistic Resistance and 101 Energy Resistance. So once you get that 4th rank of Armorer, you're essentially wearing a full suit of basic Combat Armor underneath whatever armor you actually happen to be wearing.

tl;dr version: once you get the ability to upgrade clothing from the Railroad, the Vault Suit stops being the best under-armor. Switch to one of the following at the first opportunity:

Army Fatigues

Baseball Uniform

Dirty Army Fatigues

Green Shirt and Combat Boots

Military Fatigues

Minuteman Outfit

Tattered Rags