I think carbon parts are generally more expensive to their aluminum counterparts.
I'm perplexed because aluminum version is cheaper and weighs less too? What am I missing?
Example:
Profile design sonic ergo 35a costs $160 and weighs 559g
https://profile-design.com/collections/aluminium-aerobars/products/sonic-ergo-35a
Profile design supersonic ergo 35c costs $245 and weighs 673g.
https://profile-design.com/collections/carbon-aerobars/products/supersonic-ergo-35c
You may have been missing the different armrest offset ranges. I have relatively little experience with clip on aero bars, but I think offset, which is Profile’s term, relates to fore-aft adjustment. If I’m interpreting correctly, the carbon bars’ armrests can be positioned further aft. Perhaps this is a desirable characteristic that people might be willing to pay a premium for, but perhaps it is not (see Chris H's comment on this answer).
I suspect there’s nothing intrinsic about material properties that enables the carbon bars to withstand that more rearward position where the aluminum ones would fail. I lack experience with this type of product, so I could be wrong. But basically, this specification difference could just be pure marketing.
In general, carbon does dampen vibrations better than aluminum, and cyclists are willing to pay premiums for carbon handlebars. However, I am not sure how this plays in with clip on aero bars, which are clamped to your existing handlebars.
Last, as a side note, manufacturers’ specified weights can be off. Usually the mean actual weights are higher than what they specify. Also, there will always be some variance around the weight, I.e. each aerobar will vary randomly in weight. That variance can also be systematic. For example, aluminum rims are extruded through dies. Because of entropy, the dies wear slightly through their life, so the mean weight of the rims will creep up over time.
Your title implies a question about which to get. You could attempt to confirm this with a shop that deals with a lot of triathletes. They’ll be more familiar with this product as well as the specific brand than I am. If there’s a shop that has a lot of long distance riders, including randonneurs or bikepackers, they might also be familiar with clip on aero bars as many members of those community use clip on aero bars. At any rate, if the specs on Profile's website are factually correct, it is not clear if there are functional advantages to the carbon bars.
In terms of economics, raw Aluminium costs more than Carbon Fibre and its extras. Where the equation flips is when you work in labour costs.
Aluminium parts can be molded or hydroformed in a mechanical process, that requires an accurate die or mold. Once that is made, subsequent part cost drops off rapidly.
By comparison, every carbon part is cut from flat cloth and then "assembled" and a lot of the assembly process is by hand. This layup is labour and time intensive, and that's why carbon fibre parts cost more.
As for weight, an aluminium part has a density of around 2.7 g/cm^3 where carbon fibre is 1.75~2.0 g/cm^3 So for an otherwise identical shaped part to weigh more in CF than AL, there must be about half as much more material inside. For your aerobars, it would be interesting to measure the wall thickness both at the ends and right in the middle. I'd expect the CF bar to he twice as thick as the AL.
Why? I'm no metallurgist, but my understanding is that when Carbon fibre fails, it fails very quickly. And one mitigation is to simply beef up the parts. For aluminium it fails faster than steel but slower than CF. That is, a crack forms and propagates over some time. On the other end is steel, where the last 10% of a part might hang on for a strangely long time before failing, so you have the chance to pick it up on maintenance or inspection.
