As @KMSTR said, one of the most important aspects of vector graphics is the fact that they are resolution-independent. That is, you can create your vector without having to take the ultimate pixel dimensions in mind from the very start of the project. In an increasingly mobile and retina-populated world, this is a very important asset.
It also prevents you from having to create each and every logo you design in a huge (both in digital and optical dimensions) raster file, in case it has to be printed on a scaffolding ad or large billboard. Such a raster file is going to be at least several hundred MBs (uncompressed), while a typical vector file for a logo will be several MBs, tops. This relatively small vector file can serve as a source for any raster file of any size.
Then there's the ease of editing. Especially with advanced options in vector software, it's easy to change a minor detail about a design without having to manually change pixels. Say I designed a web button, but I turn out to have made a typo -- a character is missing. In raster, I'll have to resize the button and the type, manually editing many pixels in the process. In a vector program, I'll edit the text, move a few anchor points around, and my button is ready.
Don't forget that lots of typical raster programs already incorporate many a vector-like function. Type is typically (no pun intended) rendered as vectors, unless you specify otherwise. Shape layers and vector masks are other Photoshop examples just off the top of my head.