Draco and Lucius get super angry and over dramatic about Buckbeak hurting Draco’s arm and tried to get Hagrid fired and Buckbeak killed as punishment in Harry Potter and the Prisoner of Azkaban. They did nothing about Mad-Eye Moody (Barty Crouch Jr.) transfiguring Draco into a ferret as a punishment for being rude to Harry in Harry Potter and the Goblet of Fire. I understand that they might be in on the plan or scared of Mad-Eye (the real one) but is there a canonical reason for the different reactions to these events?
Because Draco didn't tell him.
Draco threatened to tell Lucius, to which Moody replied:
"Oh yeah?" said Moody quietly, limping forward a few steps, the dull clunk of his wooden leg echoing around the hall. "Well, I know your father of old, boy... You tell him Moody's keeping a close eye on his son... you tell him that from me..."
-Harry Potter and the Goblet of Fire, Chapter 13: Mad-Eye Moody
Remember, in the book, only Voldemort (and maybe Wormtail) knows who Moody really is, so the Malfoys would have every right to be a bit nervous that Draco (and therefore Lucius) had drawn the attention of one of the greatest Aurors. In this case, it seems like Draco just kept his mouth shut not to further irritate Moody and put his father into danger.
Moody is effectively saying "Yeah... Go ahead and tell him. Also be sure to tell him that it was me who did it. I know who you are and I know who he is... and what he was. Go ahead and pick a fight with me."
NKCampbell's answer already addresses the second event, but doesn't touch on the first:
Why did Draco and Lucius get super angry and over dramatic about Buckbeak hurting Draco’s arm and tried to get Hagrid fired and Buckbeak killed as punishment
Hagrid is a half-giant. It's unsurprising that Lucius should seize on any excuse to get him fired.
Goblet of Fire was a pretty poor time for petty political power plays.
In The Chamber of Secrets, Lucius Malfoy makes a serious attempt to remove Dumbledore from Hogwarts, and it backfires pretty spectacularly. Dumbledore remained in power, Lucius lost his seat on the Board of Governors (and his house elf!), Lucius is tagged by Dumbledore as possessing a powerful artifact of Voldemort's (even if he can't prove it), a powerful Slytherin secret in Hogwarts is destroyed, a powerful Gryffindor secret is found, etc, etc.
All in all, things went poorly.
Hagrid, as Lucius's scapegoat of choice, played a role (albeit a small one) in this disaster. So when an opportunity for petty revenge fell in Lucius's lap, he seized it. It's not much, but it was a way for Lucius to thumb his nose at Dumbledore, so he took it.
Sirius's escape didn't bother him. I don't know if Lucius knew that Sirius was innocent, but either way there was no particular reason to expect that his target was Draco, who is really the only person at Hogwarts that Lucius cares to protect from harm. The dementors were equally unlikely to bother Draco. So Lucius was free to pursue his petty revenge without worry.
The atmosphere around The Goblet of Fire was very different.
First off, the presence of the Triwizard Tournament meant that Hogwarts was under international scrutiny. High levels of scrutiny are not a good atmosphere for petty tricks.
Furthermore, Moody's appearance at all was a power play of Dumbledore's own. You don't hire a PTSD-ridden paranoid retired special forces agent to teach high school students because you think it will give the students a nice educational year. Dumbledore bringing Moody on campus was a sign that Dumbledore was expecting trouble. When Dumbledore is expecting trouble is not a good time to be causing trouble.
Finally, the signs of Voldemort's return were all over the place. The Dark Mark at the World Cup and the strengthening of the Dark Mark on his arm would have signaled to Lucius that Voldemort was preparing a play of his own.
All in all, this seemed like an excellent year to keep your head down and not make any waves.
If you look at the different situations. There is a massive difference in control, and actual danger. To the point where, even if he did talk to his father about being turned into a ferret. I seriously doubt Moody would've gotten more than a warning.
The transfiguring into a ferret is a spell used by the teacher of the Dark Arts as punishment. Yes it might've been unorthodox but at least it was a controlled and conscious decision with minimal risk.
The situation with Buckbeak is different. This is a teacher who let a kid get hurt whilst giving his first class. And if you look at how it happened it's not actually that weird that Lucius/Malfoy wanted to get him fired.
(Emphasis mine) It happened in a flash of steely talons; Malfoy let out a high pitched scream and next moment, Hagrid was wrestling Buckbeak back into his collar as he strained to get at Malfoy, who lay curled in the grass, blood blossoming over his robes.
‘I’m dying!’ Malfoy yelled, as the class panicked. ‘I’m dying, look at me! It’s killed me!’
‘Yer not dyin’!’ said Hagrid, who had gone very white.
‘Someone help me – gotta get him outta here –’ Hermione ran to open the gate while Hagrid lifted Malfoy easily. As they passed, Harry saw that there was a long, deep gash in Malfoy’s arm; blood splattered the grass and Hagrid ran with him, up the slope towards the castle.
Very shaken, the Care of Magical Creatures class followed at a walk. The Slytherins were all shouting about Hagrid. ‘They should sack him straight away!’ said Pansy Parkinson, who was in tears.
Honestly while the situation ended up being reasonably fine, it was just a gash on his arm. But if that was a hit on his neck or if Hagrid wouldn't be able to control Buckbeak. Malfoy could've been seriously maimed or dead. That's more than enough leverage to actually try and get Hagrid fired and Buckbeak killed.
There are really good points in the previous answers and comments, but I think some are still left.
As Peter rightly explains, Lucius would be motivated by racism. But through Hagrid, he wants to attack Dumbledore similarly to what he did the previous year. And it is also very likely that he wants to get some revenge for the end of the Chamber of Secrets.
It was not established for sure whether Draco did tell his father about it. But compared to the previous case, it was a really humiliating situation for Draco. And in front of a large section of the school. And he was eventually saved by McGonagall, not a person he would be so keen to recognise. Plus it is likely that he was simply scared of Moody.
If he still went through, Lucius might have considered the following points:
- While the credibility of Hagrid was never brilliant, Moody is famous and well listened,
- He could not get to Dumbledore through Moody, and would not have that much to profit from it (generally up to that point, Lucius had always been less emotional than Draco),
- The two previous years have tarnished his own position and he might want to play cautiously until the next big organised thing. Not playing so much solo.
- It has been established (mostly in the books, though), that Lucius had a collection of illegal Dark Artifacts. He tried to get rid of them, but could not do it so easily. He most certainly did not want to attract too much attention from the Aurors, and less so to the most famous of them.
Because Lucius is a bully and a coward
Buckbeak, Hagrid and, by proxy, Dumbledore could be pursued through the law because they play by the rules.
"Mad-eye" Moody was a rogue element and potentially very dangerous - pick on him and he's likely to retaliate in ways that might not be, strictly speaking, legal.