Illustrator is vector graphics while Photoshop is bitmap graphic. There's little relation between each other.
OK. Photoshop has the pen tool and basic vector capabilities but vectors are not its forte.
Illustrator does have some nice effects for jpgs and pictures but, again, if compared to photoshop, it is limited.
Which one is the best for web design? both. Or none.
It really depends on what you want to do.
Photo manipulations, beautiful and very sophisticated gradients (no linear or regular), millions of colors, collages and more = photoshop
Logos, graphics with solid colors or "basic" gradients, creating typography, drawings with smooth lines and curves = Illustrator