These are the differences:
"That's not good"- correct
"That's not well" - weird
Normally and generally, the word "good" is used to be an adjective (describes nouns) while the the word "well" is to be an adverb (describes verbs).
I saw (verb) it well. (rather than "I saw it good").
It's a good thing (noun). (rather than "It's a well thing").
But when you are talking about "well" in the meaning of healthy then there is twist and it also describes a noun. For example: "She is well" (she is healthy) or "She is not well" (she is not healthy).
For more information, read here and here.