The STIS Deep Field (and the other ones too!) were made up of hundreds of separate exposures, which were added together after they were made. This hugely increases the amount of light you can detect.

About the Hally image, I am surprised the ESO didn't blur the image a bit; a faint smudge is much more noticeable that way. They could have kept the good resolution image and put the blurred one next to it to enhance the comet's visibility.