Should have less power draw, but the 2K cameras tend to use newer chips, which are often more efficient, and since they are normally mid range rather than budget cameras, they often have more efficient power regulation, so FHD may have lower power draw, but it is definitely not guaranteed.
Image quality is defined by the bitrate, so two cameras that have "equally low bitrate" should have the same image quality, unless one suffers from some other issue such as a poor lens quality. Using 2K gives the advantage that it can be optimised for better spatial resolution, while the FHD camera would have to use that bit of the bitrate for something else such as temporal resolution, but that doesn't give "better image quality" for the FHD, just the possibility of a different compromise if that is what the designer decided on. Overall I would expect to get better image quality from the 2K camera if both were using the same bitrate, and I could select how to make use of the bitrate myself.