Some runes don’t make sense to have always on Having a rune of disintegration tattooed on your fingertips is going to put a serious crimp in your love life. Etching the same thing on armour? Not so much.
Armour can be bought and applied post-hoc Imagine you’re the lord of the manor, and you have occasional need of a militia. You buy a hundred Helms of Not Dying Instantly to give to your militiamen as and when they join up. That gives you quite a lot of freedom to pick and choose who to and when you give power.
Some runes don’t make sense to leave in the world In the same example: you don’t want the peasantry walking around with Runes of Murdering etched on their chests (lest they get any funny ideas about the nature of feudalism), but if you can hold the Breastplates of Murdering in your armoury then you can dole out power when needed.
Armour gives flexibility Facing invisible foes? Wear the goggles of Seeing Better. Trying to kill Ents? Perhaps the Flaming Helmet Of Flammability is a better bet. You can’t swap tattoos anywhere near as easily.
Armour can take more runes This depends on your exact processes, but generally speaking you can etch more finely than you can tattoo. That lets you put more runes on the same space on a piece of armour. Not only that, but armour has both an outside (that you can cover in Runes of Beartrap) and an inside (for runes of Touchy-Feely-Keeping-You-Alive). On top of that you can add lots of cool spikes and swoosh detailing that would just be impractical and get you killed in the real world, but provide more space for additional Runes of Badassitude.
Armour can be mass produced Your standard dwarven production line can’t churn out a hundred tattoos in a day. It can churn out a hundred enchanted boots though.
Addendum thanks to Starfish Prime
Armour can be upgraded easily Suddenly realised you need an extra boost? You can get someone to file a rune off your greaves and re-etch, or even just add an extra rune. Unlike with tattoos it won’t hurt or even require you to be there! You can just pay the enchanter, leave your gear and come back later (or wait and have a goblin shine your shoes). Hell, if you really want to you can just buy a new set of whatever bit of armour you need to replace and trade the old in for scrap. It’s hard to do that with your own skin.
With some combo of these reasons I can certainly see armour sticking around, though sadly the psychic requirement means that armour being inheritable would just leave you with sad mementoes instead of your Grandad’s Armour of Kicking Ass.
Oh well.