In "Harry Potter and the Half-Blood Prince" in the end of the "That's for my father" incident with Malfoy on the Hogwarts Express, Luna Lovegood finds Harry Potter under his cloak with the help of Wrackspurts revealing glasses.
But in the books this cloaks make the wearer invisible for any charms and spells. Is this a plot inconsistency or just a screenwriter have written a bit of nonsense?
But in the books this cloaks make the wearer invisible for(from) any charms and spells.
This assumption is incorrect on your part. I would like to draw your attention to the scene right before Dumbledore is killed by Snape. Harry and Dumbledore come back to Hogwarts after destroying a Horcrux.
'Go and wake Severus,' said Dumbledore faintly but clearly. Tell him what has happened and bring him to me. Do nothing else, speak to nobody else and do not remove your Cloak. I shall wait here.'
So at this point we have established that Harry is wearing the cloak of invisibility. Now, cut to the scene where Dumbledore is disarmed.
Then, by the light of the Mark, he saw Dumbledore's wand flying in an arc over the edge of the ramparts and understood ... Dumbledore had wordlessly immobilised Harry, and the second he had taken to perform the spell had cost him the chance of defending himself.
This proves that the wearer of the invisibility cloak is not immune to spells and charms but merely invisible to the naked eye.
The way I understood the scene, Luna was able to see the wrackspurts around him. I see it being similar to how flies might be circling an invisible trash can. Luna's beliefs in such things may be considered insane by most of the wizarding community, but I guess that's just how they decided to do the scene. Therefore, I would say it's a plot inconsistency/hole.
In response to those who feel that Dumbledore's ability to immobilize Harry at the end of book 6 is evidence that the cloak does not ward against spells or magical abilities, we must remember that this is Dumbledore we're talking about. Dumbledore is considered the most powerful wizard in living memory, possibly of all time.
If that is the case, then it can simply be explained by Dumbledore's petrification spell being strong enough to pierce any protections that the invisibility cloak might have.
However, in book 7 it's also shown that Harry's invisibility cloak was resistant to spells:
Then again, In the beginning of book 6, it's shown that Draco Malfoy used Petrificus Totalus on Harry while he is invisible due to the invisibility cloak, on the train ride to Hogwarts. While Malfoy's abilities are noted to have increased significantly between books 5 and 6, it is all but guaranteed that he was still nowhere near Dumbledore's capabilities. With the events in book 7 displaying the resistance properties of the invisibility cloak more prominently than any of the previous books, I think this is just a plot hole, this time by J K Rowling. As above, a death eater's spell had no effect on the cloak and the Thief's Downfall securing Gringott's from magical intrusion is likely to be a far more powerful effect than Malfoy's Petrificus Totalus.
While the (lack of) protection of the cloak's wearer from the spells was well explained in KeyBrd's answer, the invisibility was not, so I'd like to address that.
It was established throughout the books that the cloak is an exceptional one, not losing its properties, etc. However, some devices can penetrate it. One of them is Moody's magical eye, as explained in the following excerpt from "Harry Potter and the Goblet of Fire":
Moody took a step closer to the foot of the stairs. Harry saw Moody's magical eye travel over Snape, and then, unmistakably, onto himself.
Harry's heart gave a horrible jolt. Moody could see through invisibility cloaks... he alone could see the full strangeness of the scene: Snape in his nightshirt, Filch clutching the egg, and he, Harry, trapped in the stairs behind them. Moody's lopsided gash of a mouth opened in surprise. For a few seconds, he and Harry stared straight into each other's eyes. Then Moody closed his mouth and turned his blue eye upon Snape again.
Recall that it is the eye that sees through the cloak, because - as we learn later on - this Moody is not the real Moody, but rather Barty Crouch Jr., who has no extraordinary wizarding powers (although, he could've use Homenum revelio spell, but I don't believe that he did). In the movies, that scene is not present.
The aforementioned movie and book scene in the Hagrid's cabin (from "Harry Potter and the Chamber of Secrets") is explained by J.K. Rowling:
Angela Morrissey: Why is it that Albus Dumbledore can see Harry under his invisibility cloak at certain moments? (during the series is the cloak only infallible to those who do not own a deathly hallow).
J.K. Rowling: Dumbledore, who could perform magic without needing to say the incantation aloud, was using "Homenum revelio", the human-presence-revealing spell Hermione makes use of in Deathly Hallows.
Another device that can see through the invisibility cloak is the Marauder's map, as Lupin explains in the book "Harry Potter and the Prisoner of Azkaban":
"The number of times I saw James disappearing under it...", said Lupin, waving an impatient hand again. "The point is, even if you're wearing an Invisibility Cloak, you still show up on the Marauder's Map. I watched you cross the grounds and enter Hagrid's hut. Twenty minutes later, you left Hagrid, and set off back toward the castle. But you were now accompanied by somebody else."
As the name "invisibilidy" hints: the cloak hides its wearer from the sight, but not from all the magical spells and devices.
If I remember correctly, in the books, it's Tonks who got Harry off the train. She did so with the use of a spell, the same way Malfoy petrified him. While it is completely possible to still affect the person wearing the cloak with spells, it's also feasible she could have used the glasses to "see" Harry.
The cloak is impervious to any spells cast at the cloak. However, if I know where the person with the cloak is, I can cast a spell on him. For example, in books, Malfoy uses freezing charm on Harry in the train because he knew where he was having seen his sneakers earlier. In movie, Luna saw the Wrackspurts, and not Harry himself, because Wrackspurts, like Dementors, apparently could still sense the human presence, emotions etc and gravitated towards it. So she deduced there was someone there and knowing about Harry's cloak, deduced it was him. In books, I believe Tonks, who noticed Harry hadn't left the train, used the spell to reveal hiding humans, that Hermione uses in 7th book, on each compartment as she says she knew he had the cloak. Since here the charm is not being cast at the cloak but at the human presence, it would reveal the person.
We also see that in Deathly Hallows, Death Eaters try to summon the cloak off Harry but it doesn't work because that was cast at Harry.
On the other hand, Moody's eye, Marauder's map, etc are designed to look through objects and enchantments, which means that they could see through the cloak. Hence the cloak can't be used to hide from such things.
In short, no spells would work on cloak, but you can still cast a spell on the human under it if you know where they are at.
