Why did Severus not want the best of him revealed?

In Harry Potter and the Deathly Hallows book, Severus says:

But never - never tell, Dumbledore! This must be between us! Swear it! I cannot bear... especially Potter's son... I want your word!

Dumbledore replies:

My word, Severus, that I shall never reveal the best of you?

Dumbledore asked Severus to protect Lily's son, (this is Severus's response) so i'm guessing this is the "best" of him? Instead of making a whole new question about the same subject, can anyone confirm this, if not tell me what the "best" of Severus is? He also said, "especially Potter's son," can anyone include why he said this as well in their answer?

Anyway, why did Severus not want the best of him revealed? And what is the "best" of Severus?