The primary advantage offered by clipless mountain bike shoes/pedals is that they allow for increased pedaling efficiency. Since your feet are secured to the pedals via the clipless shoe interface, you are able to pull the pedals upward and get more power out of the upstroke. Whereas with platform pedals, you are only getting full pedal power on the downstroke and allowing the momentum of the cranks to pull the pedals upward. Because you're able to spin in more continuous circles, you'll experience a higher cadence with clipless pedals. For this reason, they are an excellent upgrade for cross country riders.
Since clipless mountain bike shoes essentially make you one with the bike, you'll have tighter control of the bike and enhanced maneuvering. This can be particularly useful in competitive mountain biking where every small advantage adds up. The clipless pedals also keep your feet in the prime pedaling position, so that you don't fumble around and lose energy and time with foot slippage and poor positioning.
Another advantage of clipless mountain bike shoes and pedals is that they offer increased stability and safety. Since your feet are unable to simply bounce or slide off the pedals, as is the case with a platform pedal, you don't risk having your foot suddenly fall off and hit the ground causing a crash or injury. Another painful injury that is avoided with a clipless system is having the pedal spin around and hit your shin.
In addition to the broader advantages offered by the clipless system in general, clipless mountain biking shoes are sport-specific, which means they're designed with mountain biking performance in mind. Unlike the sneakers or hiking shoes that you may bike in with platform pedals, clipless mountain biking shoes employ a stiffer sole that provides better power and stability when pedaling.
While clipless systems offer distinct advantages, there are also some disadvantages. First, they are more expensive than a simple platform system. Unless your bike comes with clipless pedals, you'll need to purchase and install them separately and also purchase clipless shoes. Though you can learn to use clipless pedals effectively, they offer the problem of a more difficult dismount. If you crash or come to a sudden stop, you may be unable to release your shoes from the pedal and may suffer injury this way. Many clipless shoes are also stiff and uncomfortable to walk in. The decision to go clipless really depends on the rider's style and preferences.