The disc harrow lets you move a lot more quickly and doesn't require PTO power. However, the tiller gives you the option to vary the speed and depth and therefore how much vegetation gets mixed in and how thoroughly with just one pass. The tiller doesn't like rocks; a disc doesn't either, but will generally roll over them with minor dulling of the discs. The tiller will leave a more thoroughly smoothed and bare surface, while the disc will take multiple passes to get most of the vegetation buried. The disc can be used by itself or as a leveling tool after using a plow, while the tiller would usually be used by itself. You will probably need to creep along at 1mph or less with a tiller, while the disc can be used at 3-5mph if you have the power. Traction and drawbar pull are important for discing, while only PTO power matters for the tiller.
I have a disc, and for doing multiple acres at a time it is the tool to use. The tiller would be a good choice for doing garden plots or any task where you need to intensely work a small area of soil. Either one can mix in soil augments, but the tiller can probably do it more thoroughly.
One thing to note -the impressive videos you may have seen where a large farm disc turns a field of last year's crop trash into rich black dirt does work for farm discs, but only because (1) they are going over a field that has been worked with annual crops for years and doesn't have any well-rooted weeds, and (2) farm discs are bigger and heavier than what you can pull with a CUT. The smaller discs are basically the same in what they do, but don't do it as thoroughly or deeply as the big offset discs used by farmers, and they won't convert healthy grass sod into a ready-to-plant garden plot, except with multiple passes spaced over weeks.