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.