I think it's the same case as "the glass is half full" or "the glass is half empty". It's your perception of life. "Pretty good" is a point of view of an optimist, "not bad" is that of a pessimist.

Further to the earlier answers, I would add that the phrases are generally considered colloquial. Personally, I wouldn't use them in a business letter unless writing to a client that I'm quite familiar with.

"Good" and "bad" by themselves can sound emphatic. In light conversation, we really don't want to commit ourselves so strongly. So when you describe something as "pretty good", it is a positive sense. You enjoy it, or think it is beneficial, or that person is skilful: "She's pretty good at computers." You can also use this to convince someone in a friendly way: "Try this chocolate, it's pretty good." "Not Bad" is also saying something is "good" without making yourself commit heavily and therefore justify your opinion. "How was the movie?" "Not bad." So you enjoyed the movie. :) You won't use these in a debate or argument, of course. You'll use them casually, and so the conversation stays friendly and lighthearted. The important thing is not to analyse it too much. ;)

it really depends on the situation: pretty good means "fairly good". It's better than "not bad", but not as good as "Good" . In some cases "Not bad" means "really good" if it is said with a smile or in a happy tone.