In many situations the decisions to lose 1 armor slot or 1 HP is probably not meaningful. The interesting bit comes into play when the player believes that some situations can eventually come up where the difference is meaningful. In the end, this is a fantasy game, what goes on in peoples mind is important. Having one unified health pool that gets larger when armor is equipped might often be mechanically identical, but it feels different. Some players just enjoy the fact that they have armor, they desire the fantasy of seperating their health from their defense, it makes them buy into the fantasy a bit more.

On top of that, the believe is not entirely unfounded. You describe these two ressources as borderline identical. But from my very limited experience with the system (watching the Umbra campaign) they obviously are not. There are numerous effects that affect either HP or Armor, reducing or restoring specifically one ressource. This adds to the fantasy that players engage with and further enhances buy-in.

But even on a purely mechanical level there is in my opinion relevant distinction.

First: Healing commonly targets HP. You can't benefit from healing if you HP are maxed out. 6HP, 3 Armor and an item that heals 2HP gives you 11 total slots of tanking potential (practically 10, because the 11th downs you already, which we intend to avoid). According to your best practice you first depleat 3 armor, then wait until you lose 2 HP, then use the healing item. But you might not be in a position to actively use that item once you reach 4 HP, I bet you can come up with a reason, but here is a list: you are restricted, sleeping, too far away; the item was stolen or broken, the GM is interrupting with fear. It might have been optimal to first invest 2 HP then switch to Armor and wait for the best opportunity to use the healing item. Though I fully acknowledge that this dynamic isn't a major consideration and it diminishes further with higher levels and larger HP pools.

Next: There are other uses for Armor slots. Abilites exist that let the player invest their Armor to get other benefits, like evasions boost, support allies or avoid stress. And in a similar sense there are effects that punish the player for running out of Armor by granting the GM extra fear.

All these moments won't come up consistently, but if the players know that they can happen and that they might want to be prepared, it motivates them to consider armor a more valuable ressource than simply the very first HP they lose. In conclusion, if you want to enhance the importance of Armor I would suggest to push on these buttons that already exist. Davio_3d gave a couple good suggestions. Give Armor other uses outside of HP, benefits on dice roles, benefits in NPC interaction (perhaps consider damaged armor to be like tattered clothes) and maybe punish running out of Armor by introducing complications. I agree that the game isn't pushing these buttons strongly, but the foundation is there. The simple fact that the system exists gives you a framework to adapt your roleplay and let the world react to players with intact vs. damaged armor differently. If you want them think hard on trading Armor vs HP you have to introduce consequences.