It's literal. If psychology.stackexchange.com existed, they could help better than me, but basically, if Harry opens his mouth words may come out that indicate his true response to Dudley's appearance.
These words would very likely not include handsome, grown up and pride inspiring.
Given the context of how the mentioned protagonists already feel about Harry, he decides it is safest not to say anything at all. He may start off trying to say words that the other members of the household want to hear, but during any ensuing conversation, he will have to say something reflexively, and this might be closer to what he actually thinks. The expected response from the household to that would not be likely to be pleasant for Harry, so he literally is not trusting himself to speak.
Additionally, he has concerns about breaking into uncontrollable laughter. Devoting effort to talking would make it more difficult to prevent this from happening. I would have picked this as the reason, but in that case a better lead in sentence would be 'Harry didn't trust himself to open his mouth.', so I feel the speaking is more related to the fact that Vernon and Petunia have both uttered highly positive sentences and Harry can't carry that off. Contrast this with either of the twin Weasely brothers, for instance, who would have no problems in trusting themselves to speak.
Both phrases are valid, but they both mean slightly different things. The phrase "impossible is nothing" evolved from "nothing is impossible".
Consider the following dialogue:
Person A: We can't do X - it would be impossible!
Person B: Nothing is impossible.
In this scenario, person A claims that doing X is not possible, but person B claims that this is not true. "Nothing is impossible" is a rejection of the claim that X cannot be done, by suggesting that problems abandoned for being "impossible" are typically "hard" rather than "impossible".
The phrase "impossible is nothing" is a very modern extension of this phrase into a boast, as shown below:
Person B: Nothing is impossible.
Person C: For me, impossible is nothing.
In this scenario, person B claims as before that many things people think are impossible are merely hard. But person C is saying that such problems are nothing - that is, trivial - for him.
So in summary, "Nothing is impossible" is a motivational phrase rejecting claims that something is "impossible" by claiming that it is only "difficult" rather than "impossible".
But "Impossible is nothing" is a boast - a claim that the speaker is able to easily perform feats that other people would think impossible to achieve.
Best Answer
Conceptually, there's not much difference at all.
However, the phrase burst into tears conveys the notion of a sudden outburst of crying. It's possible to cry a lot without bursting into tears. You could weep more gradually, sniffling, sobbing, and crying one tear at a time over an extended time span. For example, a chef might "cry a lot" while slicing onions, without ever "bursting into tears."
Bursting into tears is an expression that usually describes someone who breaks into sudden weeping after being overcome by a strong emotion, such as joy or grief.
It's also possible to burst into tears without crying a lot. For example, after a mining accident, the father of a rescued coal miner might "burst into tears" upon hearing the news of his son's rescue, but then quickly regain his composure and smile in a huge sigh of relief, hugging his neighbors and other supporters.