double majors usually exist in more flexible degrees like arts, science, etc, not usually for more "strict" programs like engineering or health courses. essentially, where you would be picking elective units or a minor in your degree, that's replaced with units from your second major, so your classes are basically more restricted to the majors you picked (less electives to pick from a variety of different majors/fields). the simplest way to think about it is that you'll be taking all the required classes to graduate with one major, and then all the required classes to graduate with the second rather than flexible classes you would otherwise pick. for example if you majored in chem and math, you'd study all of the core chemistry units (organic, physical, inorganic etc), and then all of the core math units (analysis, linear algebra, etc), and then perhaps some other elective units left over. it also varies based on the uni - pretty sure at unsw if u did a double major then the required units to take would be halved to "split" the load of each major (eg if each major was 90 credits then you'd only have to take 45 credits in each major instead), leaving all the electives free, whereas at usyd your electives are used up (not entirely sure on this though).
double degrees basically combine two different "areas" of coursework together (different faculties basically), both outside of the scope of each other. for example if you wanted to do economics and bio as a double major, it wouldn't be possible since a science degree doens't offer a major in economics and vice versa for a business degree. however if you take on a business and science double degree, then you're able to complete coursework in these two different areas, and at the end you'd graduate with two bachelor degrees rather than one. usually a double degree would take ~1-2 years extra, since you have to basically complete the work required for two degrees - although some requirements like the amount of electives would be removed from each, so you won't have to take the 6 years that might be required (assuming each degree individually takes 3 years). so it's advantageous if you want to be skilled in multiple different areas (again when i say areas i pretty much mean different faculties - science, arts, business, law etc), without having to take multiple bachelor degrees separately, although this obviously comes at the cost of some extra time (not really that much though). it does make you more multidisciplinary which can be good in a lot of cases, or useless in others (some people only really end up "needing" one of their degrees but of course it's good to have lots of skills).
if i had to sum it up into one sentence - double major is exploring two different fields in one degree/area of learning eg bio and physics under science, history and english under arts etc, wheras double degree is exploring two different fields in two degrees/areas of learning eg bio and eco under science and business, history and law under arts and law etc.