The rules for the Flak Coat and the Flak Jacket in the core rulebook don't compare.

Now after the Errata the Coat covers all locations with 3 AP, and yet is still cheaper (80) than the Jacket who covers not the head (100).

Why would anyone want a Flak Jacket when the Flak Coat is superior yet cheaper?

its easier to hide a jacket thean a coat, and look less armed.

from the errata:

The Flak Cloak on page 145 should cover the following
locations: Head, Arms, Legs, Body.

The Mesh Combat Cloak on page 145 should cover the
following locations: Head, Arms, Legs, Body.

I'm going to go with Sarius. Stalking around in a fully body armoured cloak will draw a bit more attention then a flak jacket would. It'd also be alot easier to yank off someone's back and (at least I'd say) you wouldn't be able to use one with any other sort of cloak. On a purely, RAW, mechanical basis though yes, the flak jacket is obsolete, save that it's slightly more common than the flak cloak.

I've also just noticed you said coat instead of cloak.

I've put it in my game that the Cloak is heavily armoured and decked with flakboard plates everywhere, very obvious, meanwhile the Jacket isn't so much as so.

The Laughing God said:

Coat folks, not Cloak.

But the premise still holds that a Flak Coat is a rather obvious piece where a Jacket could be concealed under a coat. Keep in mind that this concealment would not hold under physical examination, and maybe not with observation from 10m or less since the armouring would bluk out the wearer a bit.

There's no such thing as a Flak Coat. There's a Light Flak Coat which covers the Arms, Body and Legs and provides 2 AP, but from the stats being talked about it's clearly the Cloak that's in question.

That said, the Light Flak Coat would probably be a good bit of subtle armour. The Light bit implies that it's not obviously armoured so you can probably get away with people thinking it's just a heavy coat.