We have very little to go on, canonically; there’s just the one scene with Odin in the vault. From that, I think, we’re supposed to gather that child!Thor was confident, rambunctious, and belligerent, while child!Loki was more hesitant and anxious – and especially anxious to keep up with his big brother and gain his father’s approval. This is all very much in keeping with the way Thor and Loki are portrayed as young adults early in the first Thor movie; I think we’re supposed to see the seeds of their adult characteristics in their behavior as children.

I also have some headcanons about Thor and Loki as children, one of which I put into my Thor/Loki fic Starting Over (there’s nothing sexual or romantic in this passage, though, so it’s safe for non-shippers):

Loki had been sensitive as a child, prone to cry at what others might consider small hurts or slights or disappointments. When he had grown old enough to join the other noble children of the palace at play, they mocked him for it, called him weak, babyish, womanish, unmanly. Even before adolescence, Loki had learned to keep his tears—and along with them, it seemed, his genuine smiles—to himself. Thor had grown up to be free with his emotions, to weep unabashedly for fallen comrades or lost loves, even at a bard’s haunting rendition of a tragic tale. But Loki never unlearned the lesson that vulnerability was weakness, that tears were a source of shame. He wept in the presence of others only when his anguish was so great that he could not hold it back, and even then would try to hide his tears, to turn aside or cover his face or find a way to escape.

So when Thor saw Loki crying openly, he felt like a child again, watching his little brother fall and skin his knee when their mother was not there to dry his tears and kiss it better. Loki’s wounds were too deep to salve with such simple comfort, but Thor could think of nothing to do but to fold his brother in his arms and let Loki cry on his shoulder.

That’s meant to account for the fact that Loki often seems to be on the edge of tears, but only spills over in moments of real crisis/confrontation, and that he’s basically a volcano of bottled-up emotions (resentment, jealousy, self-doubt, self-loathing) that erupts into full-fledged psychosis with the disastrous revelations of the first Thor film. I mean, Thor has some anger management problems, which is only to be expected in a patriarchal warrior culture, but otherwise seems to have a reasonably healthy relationship with his emotions. Loki is a poster child for long-term untreated mental illness.