What are the advantages and disadvantages of using a thinner thickness of hardwood for stairing/flooring?
Most stair treads are a true 1" thick also known as 5/4" milled. Anything less than that might sag when you walk on them depending on how it's framed. You can use 3/4" thick boards for the risers. 

Flooring can be all sizes for thickness depending on the application and manufacturer because you have a subfloor to attach it to. Thinner floors make transitions from a room with the flooring to a room without it a little less drastic.