Animagi keep their human minds and can undo the change themselves.
Dumbledore says, in explaining the story of Babbitty Rabbitty, that Animagi keep their human memory and intellect, while wizards who Transfigured themself into an animal become that animal mentally as well, and require another wizard to turn them back into a human.
“Animagi do not retain the power of human speech while in their animal form, although they keep all their human thinking and reasoning powers. This, as every schoolchild knows, is the fundamental difference between being an Animagus and Transfiguring oneself into an animal. In the case of the latter, one would become the animal entirely, with the consequence that one would know no magic, be unaware that one had ever been a wizard, and would need somebody else to Transfigure one back to one’s original form.” - The Tales of Beedle the Bard
Transfiguring into an animal would be quite risky, since unless someone knew they were doing it, the Transfigured wizard would have no way of undoing it. Worse, since they wouldn't remember they were a wizard, they wouldn't know to try to find a wizard and attempt to communicate to them what happened.
Once a wizard becomes an Animagus, it seems to be easier for them to transform.
“He hardly heard what Professor McGonagall was telling them about Animagi (wizards who could transform at will into animals), and wasn’t even watching when she transformed herself in front of their eyes into a tabby cat with spectacle markings around her eyes. ‘Really, what has got into you all today?’ said Professor McGonagall, turning back into herself with a faint pop, and staring around at them all. ‘Not that it matters, but that’s the first time my transformation’s not got applause from a class.” - Harry Potter and the Prisoner of Azkaban, Chapter 6 (Talons and Tea Leaves)
Becoming an Animagus seems to be far more complicated than most Transfiguration.
Dumbledore, in the same commentary in The Tales of Beedle the Bard, says that becoming an Animagus is very difficult, and since it has such limited use, very few would be willing to spend the time learning it, since its main use is disguise or concealment.
“Achieving perfect, spontaneous human-to-animal transformation requires much study and practice, and many witches and wizards consider that their time might be better employed in other ways. Certainly the application of such a talent is limited unless one has a great need of disguise or concealment. It is for this reason that the Ministry of Magic has insisted upon a register of Animagi, for there can be no doubt that this kind of magic is of greatest use to those engaged in surreptitious, covert, or even criminal activity.” The Tales of Beedle the Bard
It took the Marauders three years to do it, when they chose to become Animagi to keep Lupin company during the full moon.
“Yes, indeed,’ said Lupin. ‘It took them the best part of three years to work out how to do it. Your father and Sirius here were the cleverest students in the school, and lucky they were, because the Animagus transformation can go horribly wrong – one reason the Ministry keeps a close watch on those attempting to do it. Peter needed all the help he could get from James and Sirius. Finally, in our fifth year, they managed it. They could each turn into a different animal at will.” - Harry Potter and the Prisoner of Azkaban, Chapter 18 (Moony, Wormtail, Padfoot and Prongs)
Krum, a fairly competent and intelligent wizard, considered Transfiguring himself into a shark a feasible idea for breathing in the lake during the Second Task, so although it did end up going wrong, it was at least conceivable that he could learn to successfully do it between the time he learned about the Second Task and when he'd actually have to do it. However, Hermione tells Harry and Ron that becoming an Animagus wouldn't be a feasible solution for the Second Task, since it would take years to do.
“I know what I should have done,’ said Harry, resting, face down, on Saucy Tricks for Tricky Sorts. ‘I should’ve learnt to be an Animagus like Sirius.’
‘Yeah, you could’ve turned into a goldfish any time you wanted!’ said Ron.
‘Or a frog,’ yawned Harry. He was exhausted.
‘It takes years to become an Animagus, and then you have to register yourself and everything,’ said Hermione vaguely, now squinting down the index of Weird Wizarding Dilemmas and Their Solutions. ‘Professor McGonagall told us, remember … you’ve got to register yourself with the Improper Use of Magic Office … what animal you become, and your markings, so you can’t abuse it …” - Harry Potter and the Goblet of Fire, Chapter 26 (The Second Task)
From "Short Stories from Hogwarts of Heroism, Hardship and Dangerous Hobbies":
Information in "Short Stories from Hogwarts of Heroism, Hardship and Dangerous Hobbies", one of the e-books sold on Pottermore, supports the idea that transformation into an Animagus form, once mastered, is much easier than Transfiguring yourself into an animal by other means.
To return to a human form, visualise your human self as clearly as you can. This should be sufficient, but do not panic if the transformation does not occur immediately. With practice, you will be able to slip in and out of your animal form at will, simply by visualising the creature. Advanced Animagi can transform without wands. - Short Stories from Hogwarts of Heroism, Hardship and Dangerous Hobbies (ebook for sale on Pottermore)
From Pottermore:
There's a feature on Pottermore on Animagi that goes into greater detail on them. The information below is all from that feature.
You can't choose your Animagus animal.
When a wizard becomes an Animagus, apparently they can't choose the animal they will be able to turn into. So for example, Krum couldn't have chosen to become a shark Animagus to get through the Second Task, because he wouldn't have been sure that he'd turn into an animal that could breathe underwater. However, with non-Animagus Transfiguration, he could specifically intend to become a shark, or another animal that could breathe underwater.
Becoming an Animagus only lets you become one animal, Transfiguring isn't limited to one.
Once you become an Animagus, the Animagus transformation only lets you become that one specific animal. An Animagus can still presumably use other forms of Transfiguration to turn into a different animal.
In addition, becoming an Animagus is a permanent thing, like becoming a werewolf except far more controllable, while Transfiguration into an animal is more comparable to a spell. From the Pottermore writing on Animagi:
For example: if Viktor Krum can transfigure his head into a shark’s, like he does in Goblet of Fire, that doesn’t mean he is an Animagus. Transfiguration is a form of spell-casting, whereas becoming an Animagus is forever. Sorry, Krum.
Pottermore: Everything you didn’t know about Animagi
