Why is the index of refraction important for light passing between two materials?

It's important information as it tells you how fast light travels through the material. This can give you an insight into how the angle of refraction of the light is related to the angle of incidence at the border of the materials - this relation is called Snell's Law. Knowing the various angles light is refracted through can also be used to work out other things like the critical angle of a material (the smallest angle in which light is totally internally reflected, i.e. none is refracted.) This information is useful for fibre optic cables used in internet and tv connections.