Mostly I believe the difference is in the sleeve detail with an Argyle you have an upturned cuff with 3 buttons, the Braemar you don't
to illustrate

the Argyle sleeve:


the Braemar sleeve:


Here are some descriptions from kiltstore:

Argyll Jackets
The Argyll day/evening jacket is versatile,altering its formality with accessories. Suitable for all occasions, from dinners and Burns Suppers, balls and ceilidhs, to weddings, graduations, Highland games or simply as general day wear.

Braemar Jackets
Similar to the Argyll jacket, but with more formal Prince Charlie style cuffs, the versatile Braemar can alter its formality by changing accessories. Suitable for all occasions, from dinners and dances, to graduations and weddings, or simply as smart day wear.