Okay, these are mostly nitpicks, but they are discrepancies nonetheless. I’ve found a handful between Quidditch, Beasts and the main books, but none based on Tales. Apparently JK Rowling didn’t start writing Tales until after all seven main books were finished, so it seems reasonable that there are less discrepancies there.
In Philosopher's Stone, both Hermione and Harry read a copy of Quidditch:
At breakfast on Thursday she [Hermione] bored them all stupid with flying tips she’d got out of a library book called Quidditch through the Ages. (Chapter 9)
She had also lent him [Harry] Quidditch through the Ages, which turned out to be a very interesting read. (Chapter 11)
Since Philosopher’s Stone was set in 1991–2, this means that the book can’t have been written after then. And yet it contains two references to events that take place after 1992:
The most successful Japanese team, the Toyohashi Tengu, narrowly missed a win over Lithuania's Gorodok Gargoyles in 1994.
Two teams have recently broken through at international level: The Sweetwater All-Stars from Texas, who gained a well-deserved win over the Quiberon Quafflepunchers in 1993 after a thrilling five-day match; and the Fitchburg Finches […].
Okay, so perhaps the copy that JK Rowling published was a newer version of the book, published after 1994? Then we have another problem: it doesn't keep up with recent developments in broom technology. Here's a passage about the Nimbus broom company:
The Nimbus immediately became the broom preferred by professional Quidditch teams across Europe, and the subsequent models (1001, 1500, and 1700) have kept the Nimbus Racing Broom Company at the top of the field.
There's no mention of the Nimbus 2000 or 2001. Even more notably, there's nothing the Firebolt, which was released in 1993 and part of the World Cup in 1994. Such a revolutionary broom, yet it doesn't warrant a mention? If the author updated the passages on international Quidditch teams, then the section on brooms should have been updated as well.
Whatever the case, something smells fishy.
How many World Cups?
In Quidditch, we learn about the first Quidditch World Cup, and that it runs on a four-year cycle (analogous to many Muggle-world sporting events):
The year 1473 saw the first ever Quidditch World Cup, though the nations represented were all European. […]
The World Cup has since been held every four years, though it was not until the seventeenth century that non- European teams turned up to compete.
Except in Goblet of Fire, we get a number for the World Cup which takes place in 1994:
Ludo whipped out his wand, directed it at his own throat, and said “Sonorus!” and then spoke over the roar of sound that was now filling the packed stadium; his voice echoed over them, booming into every corner of the stands.
“Ladies and gentlemen... welcome! Welcome to the final of the four hundred and twenty-second Quidditch World Cup!”
422 World Cups spaced at four year intervals takes you comfortably into the third Millenium. To squeeze that many competitions into that space of time would mean you were having one almost every year since the competition began.
Consider as well that it wouldn't always have run: for example, during the wars with Grindelwald or Voldemort, and these numbers become even more implausible. But I would think any such event would warrant a mention, because it would have to be fairly significant. In particular, the Death Eater riots at the 1994 World Cup aren’t mentioned at all, and those would have been significant when the book was last updated. (Particularly given the events at the Triwizard Tournament later that same year.)
When Ollivander explains wandmaking to Harry in Philosopher's Stone, he says:
We use unicorn hairs, phoenix tail feathers and the heartstrings of dragons.
The entries for unicorns and phoenixes in Beasts make no mention of this property. This would be fine – perhaps the author just doesn't consider wandmaking important (perhaps fair, if this is for Care of Magical Creatures) – but this passage from Beasts puts a hole in that theory:
Dragon hide, blood, heart, liver and horn all have highly magical properties, but dragon eggs are defined as Class A Non-Tradeable Goods.
The book doesn't go into further detail as to what these properties are, or why they're useful. I find it surprising that if the author was listing useful parts of dragons, that heartstring wouldn't make the list.1
Other notable omissions are Hinkypunks, Boggarts and Dementors, all of which seem like something that would be worth including.
1 So perhaps the use of dragon heartstring is a relatively new development? Nope. An example: we know from Pottermore that McGonagall has a dragon heartstring wand, which would have been issued in c.1945. According to the introduction, Beasts was first published in 1927, and has been through 51 editions. Surely the use of dragon heartstring as a wand core would be known before the latest edition.
These are instances where the two sources are slightly unclear, rather than explicitly in opposition to each other.
As Snape marks an essay in Prisoner of Azkaban:
“That is incorrect, the kappa is more commonly found in Mongolia…”
However, Beasts begs to differ:
The Kappa is a Japanese water demon that inhabits shallow ponds and rivers.
To be fair, this isn't really a discrepancy: as the handwritten annotation points out, “Snape hasn't read this either”.
There's a slight discrepancy over the definition of a Red Cap. From Prisoner of Azkaban:
After boggarts, they studied Red Caps, nasty little goblin like creatures that lurked wherever there had been bloodshed.
While Beasts has the following definition:
Red cap: these dwarf-like creatures live in holes on battlegrounds or whenever human blood has been spilled.
I don't know if the difference between "dwarf-like" and "goblin like" is significant, but it seems a little sloppy.
An image on JK Rowling's website shows pages from the book Dragon Breeding for Pleasure and Profit, which Hagrid checks out of the library in Philosopher's Stone to identify his dragon egg. Here's part of the image:
Here we see the names of two breeds of dragon: the Portuguese Long-Snout, and the Catalonian Fireball. Neither of these get a mention in Beasts, which asserts there are ten breeds of dragon (and then lists all ten).
Now, Beasts does hedge its bets: it acknowledges the existence of "rare hybrids", of which these two might be instances. But if they're significant enough to get a name, then I think they count as a proper breed.