Snape knows that Harry can DESTROY Voldemort one day. That is why Snape protects him. Also, Lily was not killed for saving her own life, but SACRIFICING for her son. It means that she wanted to protect her son and Snape for her sake, protects her son.
Why did Snape protect Harry Ron and Hermione from Lupin?
Snape is deeply conflicted about his feelings towards the son of his school days chief tormenter on one side and the unrequited love of his life on he other. He wants to keep Harry safe for Lily's memory but sees a lot of James in him at the same time. She was the reason he left Voldemort.
He was possibly the only Death Eater to conjure a Patronus.
As far as readers and viewers know, Snape was the only Death Eater to cast a Patronus spell throughout the series. Snape anonymously sent a silver doe into the Forest of Dean to help Harry with his mission in the final book/movie.
Now, as firmly established, Snape was not the greatest fan of Harry, but that didn't mean that he ever stopped loving Lily. Dumbledore was surprised that Snape seemed to care for the boy. With a swish of his wand, Snape conjured up a Patronus) – Lily's Patronus, a doe.
Severus Snape did not show “love” for Harry before he died. He loved Lily, not Harry. He just showed him that (his love for Lily) and the fact that he switched sides for her by giving him his memories. Wanting to look into Harry's eyes was about Lily and not him loving Harry.
The acclaimed author explained that Harry paid tribute to Snape because of “forgiveness and gratitude.” Though Snape bullied Harry, he also saved him. “Harry hoped in his heart that he too would be forgiven.
Voldemort in turn knew the first half of the prophecy from Snape. He knew from his own knowledge of the wizarding world and his spies (presumably) that the prophecy could refer to either the Potters or the Longbottoms, and he knew from Snape that Dumbledore knew this too (if not more).
Pretty sure Voldemort would have known all the names of the pure blood families and the name "Snape" isn't among them, 'cause Snape got it from his muggle father.. Even if he did, I think that Voldemort turned a blind eye to half-blood Death Eaters, considering that he was a half-blood himself.
Snape's loyalty to Voldemort was never as deep as his loyalty to Lily. Since Snape was always primarily on Lily's side, Voldemort's ambitions to harm Lily drew a line in the sand that Snape could not cross. He would certainly never return his loyalty to the one that murdered Lily.
Why didnt Harry get in trouble for attacking Snape?
Moreover, by the time Harry and others gained consciousness in the hospital wing, they were already surrounded by other teachers and the Minister, who were thinking Snape was the one who saved them. So it's also a matter of personal reputation for him, which was why he didn't punish them outright.
Snape according to J.K. Rowling did have some affections towards harry if it was as strong as love we do not know, but he did start protecting harry only for lily's sake but by the end truly came to like and admire harry.
Snape is good: He's had every chance to kill Harry during his six years in Hogwarts and hasn't done so, and instead he's helped Harry stay alive through his lessons and his active participation in fighting dark forces.
According to some readers, Snape is simply evil, and has been working at Hogwarts for many years on an agenda of his own. When Voldemort finally returns, Snape turns at last, after being assured that his master is powerful enough to protect him.
One explanation would be that Slytherin seems to be kind of aristocracy-oriented house while his family was poor. So for him being in Slytherin might look like joining the club of the rich/important guys. You should compare the 11 year old Snape with these qualities, not the 38 one.
While most people in the universe thought he was a traitor who killed Hogwarts headmaster Albus Dumbledore (spoiler alert) and betrayed the Order, we find out he was just acting on the orders of Dumbledore and that this was an elaborate plan.
Knowing that Snape has been on the Dumbledore's side for all seven of the Harry Potter books reveals him as an extremely brave man and a truly skilled spy. Snape has had to face and deceive the terrifying Lord Voldemort constantly.
Having an abusive, muggle father would also have shaped Snape's opinion of muggles from an early age. This was an area that Snape would actually empathize with Voldemort, who also had negative experiences with muggles, thus rallying Snape to his cause.
Phineas Nigellus's portrait inside it overheard Hermione and reported that to Snape. Snape thus knows that Harry and Hermione are in the Forest of Dean, though not exactly where; the Forest of Dean is a largish place, several wooded sections totaling about 40 square miles.