It seems to me that the comments and answers so far have missed (or not quite made explicit) that there are two ways to read the question:
(a) Are there practical advantages to knowing richer classical logical systems (that is, systems which have some simplish classical logic as a subset)?
(b) Are there practical advantages to knowing alternative logical systems (that is, systems which in some way deviate from classical logic)?
The two answers already given have covered the practical advantages common to both pretty well, so I'll stick to the advantages specific to these senses. For (a), the advantages come in terms of ability to express and analyse more complex propositions. If you're stuck in propositional logic you can't even analyse "Socrates is a man" to anything more deep than "P", which isn't especially useful for constructing arguments. If you have a rich theory that, for instance, allows you to formulate inference rules for "K(x,p,t) -> K(x,K(x,p,t),t)" to mean "if x knows p at time t then x knows that x knows p at time t", that allows you to do lots of philosophy to a much higher level of rigour, which is a Good Thing. If that doesn't sound practical enough, think about things like rules of inference for things like modal logic, enabling us to rigorously say things like "possibly p iff not necessarily not p". I think that's certainly as practical as 'basic' logic is.
Sense (b) means dealing with stuff like quantum logic, intuitionist logic, fuzzy logic and more. Here you have some of the advantages of additional analytic power as in (a), but what's going to be more important is that seemingly viable alternative logics allow us to properly tackle questions about how fundamental logic is - do we believe in deductive logic because we just can't reason any other way, or because we're stuck in a habit, or it's a fundamental fact about the universe? Quantum theory gives rise to quantum logic, which sometimes disagrees with classical logic. So what are we supposed to do? If you accept the intuitionist thesis, a huge quantity of modern mathematics is junk. That kind of thing doesn't have practical implications in that philosophers are unlikely to get the world world to change its practices, but it is pretty significant in that it casts doubt (and light) on some of our most fundamental reasoning strategies.