Any lens that is truely designed for distant subjects must be at least somewhat asymetrical. There is nothing fundamental that makes symetrical or asymetrical designs better than the other. This reduces flare and increases contrast. There are actually three types, non-coated, single coated and multicoated. Single coating was first developed in 1936 and most lenses made since WWII have been at least single coated.
More complex designs can decrease abberations giving a sharper lens or a lens with more coverage.
Multicoating's real contribution was to allow the 12 to 15 group zoom lenses possible. Without multicoating, these lenses would be unusably low contrast. Not very much just from the numbers. As I wrote above, more groups means more air-spaces and requires better coatings. They will be sharper lenses, especially when you can avoid diffraction effects from stopping way down.