While both show the process of natural selection, vestigial structures no longer have a working function, but are still present in an organism. An example of this in humans is the appendix. To humans, it doesn't necessarily serve any function, but it is just there as part of the human body.
On the other hand, analogous structures are similar structures in different species that perform the same function. We can think of it as wings on a moth, bird, and a fly. While the three do not share a common ancestor, the structure of the wings are said to be analogous.