You have overlooked part of the benefit of the armor. The armor provides more than +1. If a non-magic version of the armor raises your AC from 10 to 12, then the +1 version is raising your AC by a total of +3 (relative to unarmored AC 10), not by +1.
So comparing the items in a vacuum, not considering the characters receiving them, the armor is better. However, in all practical use cases, the cloak is likely to be better for the player for various reasons. The easiest justification for the previous sentence is that by the time a player could get either the cloak or the armor, the player likely already has optimal non-magic armor.
But the original justification above still works in-character, in a role playing sense. Ignore the numbers and the book mechanics for a moment, and just think about an average person in context comparing the two items: a suit of armor and a cloak, both being magical and having similar magical properties.
PC: Hi, I'd like to buy that +1 chain mail on the mannequin in your front window.
NPC: That'll be $3000.
PC: But the guy down the road is selling a cloak that will raise my AC to the same amount for one-sixth that price. I'll buy it, but only if you price match for the item I would get the same benefit from.
NPC: I got a dual degree in blacksmithing/magic-crafting, and I put my blood, sweat and tears into that masterwork piece of chain mail before giving it a +1! All that guy down the road did was bum a cloak off of someone and give it a +1 with his single-degree in magic-crafting because he was too busy drinking at frat parties!
PC: Sorry for your personal issues, but I already have a normal chain-mail, so buying your +1 chain mail is not going to benefit me more than the cloak.
NPC: Do I look like I care what you already have? What does that have to do with what I'm selling? If you already had a +1 chain mail, would you expect me to give you mine for free because it wouldn't benefit you at all? Get out of my store before I tell the village idiot we've hired a replacement for him!
PC goes down the road and buys the cloak for one-sixth the price
[Your AC has increased by 1]
Other NPC: Thank you, have a nice day, come again!
All that explains away a small part of the discrepancy, but it still falls short.
The cloak is lighter, is useful to a wider array of character types, is quieter, does not scream "tough guy in armor coming through", etc.. Note that some of those benefits are purely RP benefits, not combat ones.
Most of the reason is just because of mistakes on the game creators' part, as per the answer by @Tiggerous.