What's the difference between a "pistol brace" and a "stock"? Don't they both go into your shoulder to stabilise the weapon?

There's no legal definition per Congress. Generally speaking, braces are intended to stabilize a pistol against your arm [0], whereas a rifle stock is meant to stabilize against your shoulder. However, braces can technically be "misused" such that the rear of the brace fits against the shoulder, meaning it is used as a stock. Likewise, the distinction is so small something as simple as a sling attachment to the stock could make it a brace, or an articulation that could be used as a cheek rest turn a brace into a stock, converting a pistol into a rifle or vice versa. For awhile, the only way to know the difference was for the manufacturer to submit an NFA and hope.

The ATF has been in court (and lost) quite a bit [1] over this.

[0] there's a nice picture and writeup here of a pistol brace being setup https://www.thetruthaboutguns.com/gear-review-sig-sb15-pisto...

[1] a brief rundown of the 2023-2025 legal rulings https://www.fflguard.com/atf-pistol-brace-rule/

A "pistol brace" is designed and "intended" to be braced against your forearm to stabilize the "pistol" in a way that allows you to shoot a particularly large and heavy "pistol" with one hand. The ATF said this was fine, although I think they really regret that now.