It's not generally true that paraconsistent logics reject the principle of non-contradiction. What they do reject is the principle of explosion, which says that from a contradiction anything follows (and which can be captured within proof systems by the [math]\bot[/math]-elimination rule).

The question of the utility of paraconsistent logics is tertiary to more interesting logical problems, such as the problem of the axiomatizability of various paraconsistent theories, the problem of their soundness and completeness, etc.

But since you asked specifically about their utility, I'll describe a variant of a standard scenario where they're found useful (dates back to Belnap & Anderson).

Problem. QuAC is a question-answering computer, specializing in weather forecasting. It has a number of equally reliable sources of data. When you ask QuAC a yes/no question about the weather (say, "Is A true?" where A is "It's going to rain tomorrow afternoon"), it asks the same question to its sources and collects their answers. It often happens that the sources give contradictory information, that is, one source says that A is true, the other one says that ~A is true. If QuAC were to reason classically from such an inconsistent set of answers, it would explode in the sense that it would answer "yes" to every single question (because (A & ~A) [math]\rightarrow[/math] B is classically valid for an arbitrary sentence B). Solution. Among the many solutions to this problem is to make QuAC reason not classically, but paraconsistently.