They have very similar meanings, but come are derived from entirely unrelated areas of life.
"From the ground up" is a construction metaphor, speaking of starting a project anew, beginning with the foundation.
"From scratch" is apparently a sports metaphor, although more commonly used these days as a cooking reference, as in to say that you gathered all the raw ingredients yourself (as opposed to buying a cake mix).
In the context of starting a project over, it would be appropriate to use either phrase, as in "Now we're going to have to start from scratch" or "Now we're going to have to start from the ground up", as well as many other similar metaphors (square one, from the top, etc)...