(WORNING)Why use Carbon Fiber composite filament?
Most people enjoy the semi-matte finish and chatoyant appearance that carbon composites can have, yet there seems to be some confusion about the non-aesthetic reasons for or against 3D printing with a carbon fiber composite material.
Stiffness / Brittleness
Regardless of base filament material, any carbon fiber composite should be stiffer and therefore somewhat more brittle when compared with 100% base filament. Mixing carbon fibers into plastic creates a material that is typically greater than the sum of it's parts; the semi-flexible plastic material coats and sticks to the carbon fiber strands, holding them together which helps prevent the strands from fracturing under bending stresses. This phenomenon can be roughly demonstrated using straws by holding 5 straws from a single end and bending them vs holding the same straws at both ends and trying the same bend in the middle. The straws being supported from both ends are much tougher to bend because they are being held together and able to share their strength, instead of each straw having to hold all the strain by itself.
Composite materials are a bit of a double edged sword because by adding a strengthening material that has significantly different physical properties from our base material, it creates nucleation sites for cracks to begin once the part is put under heavy stresses. The amount of additional stress the part will accept is usually significant. Unfortunately, these differences also cause a change in part failure "style"; this typically means a sharp violent break with reduced flexing when compared to the original material's properties.
Printability
The increase in overall stiffness will often reduce part warping for those warp prone materials and can enable printing in less than ideal conditions for some materials.
Glass Fiber?
Carbon Fiber is not the only composite material, although currently it is the stiffest of the composites I am aware of. Glass fiber is typically half the "composite tax" compared to carbon fiber with what I believe is 75% of the benefits and all the detriments of carbon fiber.