A computation, properly laid out, is of course a proof. However, many students, after years of multiple choice tests, have learned to take the point of view that the answer is the only thing that matters.
"Justify" can be a reminder that the problem will be graded carefully, that (contrary to their usual experience) a slapdash computation will not necessarily get full marks.
I do not think that "justify" carries any connotation of "you need only show necessity but not sufficiency."
"Prove," in a course context, can often mean that a more or less specific set of tools should be used. "Justify" has a more informal feel, but I do not think of it as carrying a lower level of precision.