I believe the existing answers are incorrect. The existing answers argue that Hagrid's wand was snapped only because the crime he got expelled for was particularly heinous, or because of some prejudice against him. I do not think this is the case. Rather, his wand was snapped because he got expelled; wand-snapping is the standard procedure for expelled students. I think this can be supported in several ways.
It does not seem to be a secret that Hagrid's wand was snapped. For example we have the comment from Mr. Ollivander in Philosopher's Stone:
"Good wand, that one. But I suppose they snapped it in half when you got expelled?" said Mr. Ollivander, suddenly stern.
Note that Mr. Ollivander supposes this, as if it was a matter of course for an expelled student's wand to be snapped. Additionally, he doesn't say that it was snapped when Hagrid was convicted for murder; he says that it was snapped when Hagrid was expelled, implying that the snapping was tied to the expulsion rather than the murder.
The quote cited in the question also supports this. Note that Hagrid is apparently concerned that people will find out that he is doing magic. If the wand-snapping was a unique punishment for his case, it is likely that most people wouldn't even know that he had received such a punishment. In fact, the reason for his expulsion itself seems to have been somewhat of a secret. So the fact that onlookers would assume that Hagrid's wand had been snapped would seem to indicate that wand-snapping is actually the standard procedure for expelled students.
Moreover, Harry explicitly associates Hagrid's not being supposed to do magic with his expulsion. Yet at this point in the story, Harry doesn't yet know why Hagrid was expelled. He certainly doesn't know that Hagrid was accused of murder. He is thus apparently assuming that wand-snapping is simply a function of being expelled.
Then we have Hagrid's description to Harry in Philosopher's Stone as to why his wand was snapped, and why he is not supposed to use magic:
"Be grateful if yeh didn't mention that ter anyone at Hogwarts", he said. "I'm — er — not supposed ter do magic, strictly speakin'. I was allowed ter do a bit ter follow yeh an' get yeh an' stuff — one o' the reasons I was keen ter take on the job —"
"Why aren't you supposed to do magic?" asked Harry.
"Oh, well — I was at Hogwarts meself but I — er — got expelled, ter tell yeh the truth. In me third year. They snapped me wand in half an' everything. But Dumbledore let me stay on as gamekeeper. Great man, Dumbledore."
Hagrid seems to link the wand-snapping to the expulsion. He makes no mention of unique circumstances. Indeed the words "an' everything" strongly indicate that the regular procedure was followed.
Finally, there is another case of expulsion in the books. Harry was expelled from Hogwarts in the beginning of Order of the Phoenix. Let's look at the letter he received:
The severity of this breach of the Decree for the Reasonable Restriction of Underage Sorcery has resulted in your expulsion from Hogwarts School of Witchcraft and Wizardry. Ministry representatives will be calling at your place of residence shortly to destroy your wand.
So Harry — who had not murdered anyone — was to have his wand destroyed. Moreover, the letter does not explain that this is a unique response particular to Harry's situation.1 It seems that the letter expects that the recipient would not be surprised at this consequence. Of course, if wand-snapping is the standard procedure for expulsions then it would probably be well-known, and that would account for why the Ministry feels no need to explain the punishment.
Of course, this leads to the question of why the wands of expelled students are destroyed, while wands do not seem to be destroyed even for committing significant crimes. We can suppose a simple answer to this. If someone is convicted of a serious crime, wand-snapping would be considered unnecessary. Criminals in Azkaban can't do magic anyway. If there is some change in circumstance that would get them released from Azkaban, that same circumstance would probably be enough of a reason to allow them to use magic. If they escape Azkaban, or are otherwise on the run from the law, snapping their wands would be pointless. You have to catch them to get hold of their wands, and even if you destroy their wands there is nothing stopping them from procuring new wands as soon as they get the chance.
The reason why an expelled student's wand is snapped, might not even be a punishment per se. It may simply be that someone without a magical education cannot be trusted with a wand. This would explain why home-schooled children would be allowed to have wands — they can be trusted with wands because they are still getting a proper magical education. A student who was originally at Hogwarts, and then expelled, will likely not have the opportunity to get a proper magical education. Indeed, in Hagrid's case he had no family to help him or otherwise provide him with an alternative magical education. In fact, we could even argue that if an expelled student would get an alternative education (e.g. by going to another school, or by having a qualified individual teach him) the Ministry would allow him to have a wand and use magic.
This would also explain why Fred and George were allowed to keep their wands and use magic. They left Hogwarts in their seventh year, and towards the end of that year itself. They already had a magical education so there was no reason to not trust them with wands. Perhaps it is thus possible that even if a student was formally expelled, his wand would not be destroyed if he was far enough into his education.
1. Harry does receive a follow-up letter which might be thought to indicate that the wand-snapping is separate from the expulsion:
Further to our letter of approximately twenty-two minutes ago, the Ministry of Magic has revised its decision to destroy your wand forthwith. You may retain your wand until your disciplinary hearing on 12th August, at which time an official decision will be taken.
Following discussions with the Headmaster of Hogwarts School of Witchcraft and Wizardry, the Ministry has agreed that the question of your expulsion will also be decided at that time. You should therefore consider yourself suspended from school pending further inquiries.
It might seem as if the letter is saying that the decision on wand-snapping will be independent of the decision on expulsion. However, it is possible that that it is actually affirming the correlation between the two. It is only because the expulsion is being reconsidered that the wand-snapping is being considered. In fact, if Harry is not expelled then he will necessarily need a wand. If Harry's expulsion is upheld then he is in the same situation as before this follow-up letter. So the only reason to reconsider the wand-snapping is that he might not actually be expelled. Thus, this would still fit with the assumption that wand-snapping is a function of expulsion.
