Based on the wide range of injuries shown the books, it looks as though mundane injuries are easy to heal, but injuries inflicted by magic (or highly powerful magical creatures) are more difficult and can leave permanent reminders of their presence. Neville's broken wrist was the kind of thing that could happen to Muggles, thus was easy to fix off-screen. Bill Weasley's werewolf wounds and Alastor Moody's injuries by Death Eaters, on the other hand, left permanent disfigurations.
There appears to be a correlation between the power and 'Darkness' of the magic inflicting the injury, and the difficulty of treating it. In the early books, and when students are cursing each other, the injuries are usually easily cured, as in the case of Gregory Goyle's boils and Hermione's outgrown teeth (in HPGOF). Later, we see injuries that are permanent and disfiguring, as the power of the magic used against protagonists increases in deadliness. Draco's Sectumsempra maiming by Harry Potter is an intermediate case, as it was a malific spell but was cast without understanding or deadly intent.
The bites from Fluffy, though from a magical creature, were probably healed with reasonable ease once Snape got treatment. George's ear, being removed by a powerful and malific Sectumsempra spell by Snape, would be much more difficult to heal. Snape himself, possessing some healing spells and great power, may have been able to do so (especially as he had been the one who created the spell). This is supported by the fact that he helped to heal Draco's Sectumsempra injuries in such a way as to leave him without mentionable scars. However, it is likely that he would only have been able to do so if George had kept his ear available for quick reattachment, as (again from Moody) we see that amputations appear to be permanent.
(edit) Upon reading the sources above, it seems that a body part severed by any kind of Dark magic is unrecoverable. But this raises another interesting point. The books have implied that Dark magic's power and presence, to some degree, depends on the malice of the wielder. A Crucio curse by Potter against Bellatrix Lestrange, for instance, is easily shrugged off, whereas her own Crucio is agonizing. We know that Snape willingly casts Dark spells, as when he used Avada Kedavra against Dumbledore...but the truth of his intent may not have been malicious enough to cause the full aftereffects of a Dark spell. The spell itself was Dark, however, and that may be enough for the amputation to be permanent, if only a trace is needed.