The biggest advantage . . .
. . . is rate of fire (Along with the other comments). A non-VLS system would require the launcher to spin back to straight-up after launching the ready-missiles, missile(s) to be rotated in the magazine carousel under the launcher, and the missile(s) elevated up from the magazine to the launcher arms . . . all before another pair of missiles could be sent on their way.
Restocking would undoubtedly be easier, since the missile boxes could be lowered down into the VLS magazines, instead of placed inside an underdeck carousel.