That's an important distinction indeed. That's actually one of the important aspects of Bill C‑40 compared to what we had before.
Currently, in order for a miscarriage of justice to be recognized and for a remedy to be ordered by the Minister of Justice, there must be a certain likelihood of miscarriage of justice. Earlier, we were talking about a threshold of 50% plus one, that is, a balance of probabilities.
In its current form, the bill actually seeks to lower the test to the level of a possibility. In the French version, proposed section 696.6 talks about cases where “une erreur judiciaire a pu être commise”, whereas in the English version, the word “may” is used. In a way, the French version talks of a reasonable possibility, which is a much lower test.
The reason the test was lowered is that it's not always easy to establish a miscarriage of justice occurred with a sufficient degree of probability. We often talk about cases that are so old that certain documents are difficult to trace, where witnesses can be hard to track down or have an imperfect recollection of events after all that time.
That explains the change somewhat.