Draco Malfoy and the rest of his family are pure-bloods and take immense pride in that. Why are pure-bloods different from half-bloods or Muggle-borns? Do they have more natural magical talent? Looking just at heredity, it would seem so, but that may not be true for wizards as they are different in many aspects.
It’s unclear if pure-bloods are generally more skilled.
There are no clearly objective statements in the Harry Potter series on whether pure-bloods are generally more skilled than wizards with less or no magical ancestry, but it is possible. Magic itself is an inherited trait, so it seems possible that having more wizards in the bloodline could mean that pure-blood wizards are generally more skilled than wizards with more Muggle ancestry.
“As intensive studies in the Department of Mysteries demonstrated as far back as 1672, wizards and witches are born, not created. While the “rogue” ability to perform magic sometimes appears in those of apparent non-magical descent (though several later studies have suggested that there will have been a witch or wizard somewhere on the family tree), Muggles cannot perform magic.”
- The Tales of Beedle the Bard
Hagrid implies that Harry is likely to be a talented wizard because his parents both were talented wizards, but it’s unclear if Hagrid is correct or if this is based on his own like for Harry.
“A wizard, o’ course,’ said Hagrid, sitting back down on the sofa, which groaned and sank even lower, ‘an’ a thumpin’ good’un, I’d say, once yeh’ve been trained up a bit. With a mum an’ dad like yours, what else would yeh be?”
- Harry Potter and the Philosopher's Stone, Chapter 4 (The Keeper of the Keys)
Much of the information in the series about whether having magical ancestry makes wizards more likely to be skilled is from very biased sources - the pure-bloods believe their all-wizard ancestry makes them more skilled, and those against the idea of pure-bloods being somehow “better” wouldn’t want to accept the possibility there might be some truth to the idea that ancestry matters. For example, Ron says it doesn’t matter at all, giving one example of an unskilled pure-blood, but he doesn’t show any evidence that it’s actually true that ancestry doesn’t generally affect magical skill.
“There are some wizards – like Malfoy’s family – who think they’re better than everyone else because they’re what people call pure-blood.’ He gave a small burp, and a single slug fell into his outstretched hand. He threw it into the basin and continued, ‘I mean, the rest of us know it doesn’t make any difference at all. “Look at Neville Longbottom – he’s pure-blood and he can hardly stand a cauldron the right way up.’
‘An’ they haven’t invented a spell our Hermione can’t do,’ said Hagrid proudly, making Hermione go a brilliant shade of magenta.”
- Harry Potter and the Chamber of Secrets, Chapter 7 (Mudbloods and Murmurs)
Similarly, Lucius Malfoy considers it shameful for his son to be beaten by someone with no magical ancestry, but he also doesn’t provide any evidence that magical ancestry increases magical talent, though he personally believes that it does.
“It’s not my fault,’ retorted Draco. ‘The teachers all have favourites, that Hermione Granger –’
‘I would have thought you’d be ashamed that a girl of no wizard family beat you in every exam,’ snapped Mr Malfoy.”
- Harry Potter and the Chamber of Secrets, Chapter 4 (At Flourish and Blotts)
Because of the inherent bias in most statements on whether pure-bloods are generally more talented than wizards with less or no magical ancestry, it’s difficult to figure out the truth. The only statement that isn’t obviously biased one way or another, and is therefore most likely to represent the objective truth, is when Slughorn says he was surprised Lily wasn’t pure-blood, since she was so skilled. This means in Slughorn’s experience, pure-bloods are more likely to be skilled in magic. He doesn’t seem to have strong opinions about blood status, and he’d have a lot of firsthand experience with how skilled wizards of various blood status are, having taught students at Hogwarts for several years.
“Your mother was Muggle-born, of course. Couldn’t believe it when I found out. Thought she must have been pure-blood, she was so good.’
‘One of my best friends is Muggle-born,’ said Harry, ‘and she’s the best in our year.’
‘Funny how that sometimes happens, isn’t it?’ said Slughorn.
‘Not really,’ said Harry coldly.
Slughorn looked down at him in surprise.
‘You mustn’t think I’m prejudiced!’ he said. ‘No, no, no! Haven’t I just said your mother was one of my all-time favourite students? And there was Dirk Cresswell in the year after her, too – now Head of the Goblin Liaison Office, of course – another Muggle-born, a very gifted student, and still gives me excellent inside information on the goings-on at Gringotts!”
- Harry Potter and the Half-Blood Prince, Chapter 4 (Horace Slughorn)
Because Slughorn wasn’t clearly biased, and taught for several years which would give him knowledge of the skill levels of many young wizards of different blood statuses, his experience that being pure-blood is correlated with having magical talent is the most likely to represent the objective truth. However, Slughorn wasn’t officially studying the correlation, and his conclusion may not be accurate. There are examples throughout the series of both skilled and unskilled wizards with every type of blood status, so no accurate conclusion can be made from evidence that’s given in the events of the series. It seems logical that having magical ancestry would make a wizard more likely to be skilled as magic is a genetic trait, but this is never proven in the series.
Pure-bloods are more likely to have magic.
However, though it’s not objectively clear how magical ancestry affects magical talent, it is clear marrying only wizards keeps the bloodline magical, and that continued intermarriage with Muggles will make descendants become Squibs and later Muggles. The Black family has kept their bloodline pure, and they’ve been consistently mostly wizards since medieval times. The Black family tree is quite full, though any Squibs are blasted off the family tree, meaning from medieval times to the present day, most of them are wizards.
“The tapestry looked immensely old; it was faded and looked as though Doxys had gnawed it in places. Nevertheless, the golden thread with which it was embroidered still glinted brightly enough to show them a sprawling family tree dating back (as far as Harry could tell) to the Middle Ages. Large words at the very top of the tapestry read
The Noble and Most Ancient House of Black
- Harry Potter and the Order of the Phoenix, Chapter 6 (The Noble and Most Ancient House of Black)
The Scourers, a group of wizard mercenaries, are able to hide themselves by marrying Muggles and weeding out any wizard children they produced, which wouldn’t be possible if they had produced magical children as consistently as wizard families - they’d have very few children that could be kept.
Several of the most notorious Scourers eluded justice. With international warrants out for their arrest, they vanished permanently into the No-Maj community. Some of them married No-Majs and founded families where magical children appear to have been winnowed out in favour of non-magical offspring, to maintain the Scourer’s cover.
- Seventeenth Century and Beyond (Pottermore)
Therefore, it’s very clear that genetics do have some effect on magic.
No. It's Explicit in Chamber of Secrets.
“It’s about the most insulting thing he could think of,” gasped Ron, coming back up. “Mudblood’s a really foul name for someone who is Muggle-born — you know, non-magic parents. There are some wizards — like Malfoy’s family — who think they’re better than everyone else because they’re what people call pure-blood.” He gave a small burp, and a single slug fell into his outstretched hand. He threw it into the basin and continued, “I mean, the rest of us know it doesn’t make any difference at all. Look at Neville Longbottom — he’s pure-blood and he can hardly stand a cauldron the right way up.” “An’ they haven’t invented a spell our Hermione can’ do,” said Hagrid proudly, making Hermione go a brilliant shade of magenta.
The natural response might be to use specific characters to refute the idea that Pure-bloods are better, and indeed Hagrid seems to do this on multiple occasions. In Chapter Five of Philosopher's Stone he says:
"Yer not from a Muggle family. If he'd known who yeh were – he's grown up knowin' yer name if his parents are wizardin' folk. You saw what everyone in the Leaky Cauldron was like when they saw yeh. Anyway, what does he know about it, some o' the best I ever saw were the only ones with magic in 'em in a long line o' Muggles – look at yer mum! Look what she had fer a sister!"
And in Chapter Seven of Chamber of Secrets:
"It's about the most insulting thing he could think of," gasped Ron, coming back up. "Mudblood's a really foul name for someone who is Muggle-born – you know, non-magic parents. There are some wizards – like Malfoy's family – who think they're better than everyone else because they're what people call pure-blood." He gave a small burp, and a single slug fell into his outstretched hand. He threw it into the basin and continued, "I mean, the rest of us know it doesn't make any difference at all. Look at Neville Longbottom he's pure-blood and he can hardly stand a cauldron the right way up."
"An' they haven't invented a spell our Hermione can' do," said Hagrid proudly, making Hermione go a brilliant shade of magenta.
However, individual examples are not necessarily valid evidence. It is theoretically possible that Pure-bloods are generally better than Half-bloods and Muggle-borns, but Hermione and Lily were exceptions. Indeed, Professor Slughorn seems to have thought it was out of the ordinary for a Muggle-born to be outstanding, and he explicitly claimed that he was not prejudiced. From Chapter Four of Half-Blood Prince (my emphasis):
"Your mother was Muggle-born, of course. Couldn’t believe it when I found out. Thought she must have been pure-blood, she was so good."
"One of my best friends is Muggle-born," said Harry, "and she’s the best in our year."
"Funny how that sometimes happens, isn’t it?" said Slughorn.
"Not really," said Harry coldly.
Slughorn looked down at him in surprise. "You mustn't think I'm prejudiced!" he said. "No, no, no! Haven't I just said your mother was one of my all-time favorite students.
If Slughorn was being truthful about not being prejudiced then it is possible that Pure-bloods generally do have some advantage, be it more raw magical power or more exposure to magic.
However, back in Chapter Eight of Philosopher's Stone Harry's assessment of his classmates indicates that there wasn't much of a significant advantage of being Pure-blood:
Harry was relieved to find that he wasn't miles behind everyone else. Lots of people had come from Muggle families and, like him, hadn't had any idea that they were witches and wizards. there was so much to learn that even people like Ron didn't have much of a head start.
This passage seems to indicate that Pure-bloods may have had a small advantage in already knowing some stuff about magic, but it implies that in the grand scheme of things such an advantage was negligible. As this is a more significant sample size than just Hermione and Lily, it may be more relevant for the purposes of this question.
Indeed, Hagrid alludes to this in Chapter Five of Philosopher's Stone:
"Don' you worry, Harry. You'll learn fast enough. Everyone starts at the beginning at Hogwarts, you'll be just fine.
He seems to believe that everyone – regardless of blood status – starts out more-or-less equal at Hogwarts.
No, "blood purity" is repeatedly demonstrated throughout the books to be nothing more than a nonsensical form of prejudice. The "pure-blood" families produce plenty of squibs and have even demonstrated inbreeding problems, and many of the most powerful and dangerous wizards shown in the books have muggle ancestry. Of the prominent witches and wizards in the series, Lord Voldemort, Albus Dumbledore, Professor McGonagall, Severus Snape and Harry Potter are "half-bloods," and Hermione Granger and Lily Evans were both muggle-born. By contrast, most of the pure-blood wizards we see, like Ron Weasley and Neville Longbottom are not particularly powerful by comparison.