Mountaineering Clothing vs hiking clothing

Though saying this sounds trite, much depends on what you mean by 'mountaineering'… there are walk-up 14rs that can be done in the summer in shorts and a t-shirt, and then there are 8000 meter peaks. Some people call both mountaineering.
You talk about "cold-weather", which for some people is 20 degrees, and other is -20. Again, there is too much variation to be overly specific.
That said, if you are above treeline in sustained cold, you will need different systems than an AT hiker… and technical climbing might call for tougher jackets for abrasion resistance against rock, ice and gear, a helmet-compatible hood, pockets for easy-access to gloves, hat, and snacks, and have long enough arms to allow for climbing. A backpacking jacket can be much simpler.
Of course, Steve House climbed Nanga Parbat in a Houdini… check out Patagonia's website for his clothing video, so much depends on the user.