In a nutshell, the difference between these services can be described as follows:
HBO Go is software that comes under your cable TV subscription with HBO included. The library of HBO Go content is limited to HBO's original TV shows and a small selection of movies.
HBO Now is a standalone streaming service and its library is absolutely the same as HBO Go’s one.
Max is also a standalone service that includes all the content available in HBO Go and HBO Now and a huge list of additional popular movies and TV shows.