Basically, a rangefinder's advantage is that it tells you exactly how close you are to the pin (or to some other object you're rangefinding), instead of approximately how close you are based on your relative distance to some other known marker, like the approach stake. With a good rangefinder, it's possible to get an accurate distance reading (usually within a yard, sometimes to tenths of a yard) on almost any object on the course, from the near to far edge of a dogleg landing, to the rocks at the near edge of a water hazard, to any stake, tree or other significant landmark through the green, and of course the pin.
In the interest of full disclosure, any device intended to be used for distance gauging, or any device incorporating this ability as a major feature of its design, is disallowed by strict rules (Rule 14-3) under penalty of disqualification. The Rules, however, also allow courses to implement a Local Rule allowing them overriding 14-3, and for casual play, virtually all courses do allow them, with many even providing such a device in the form of GPS systems in the carts.
Rangefinders have the most benefit on courses where planning your shots require you to know more than just the distance to the pin, and on courses that have more limited information along the way. If you need to know how far it is to that dogleg landing, so you can drop your tee shot there before changing direction, and there's nothing useful on the scorecard map or in your cart to help you gauge it, then a rangefinder will definitely save you a few strokes finding out by trial and error. Similarly, if the holes only have yard markings from the tees and no other distance aids through the green, and you can't rely on "dead reckoning" based on the average distances of your clubs (due to high winds, or the fact you're hitting shorter or longer than you'd expect on that day), a rangefinder can be invaluable.
On many courses I've played, however, you usually have at least an approach stake telling you you're, say, 150 yards from the edge of the green, and given where your ball actually lies relative to that stake, and how far back the pin is from the edge of the green, your distance to pin can be reckoned accurately enough to drop your ball on the green within putting distance from there. Most obstacles, similarly, are positioned such that playing over and around them is relatively natural; you can, at the very least, play it safe your first time on that course and hit away from or a little softer than you normally would, and then when you're sure your next shot will carry the hazard, you hit out longer. Your next time out, knowing how you played it last time, you might go for more club and a stronger swing to carry the tee shot over the hazard, setting up an easier GIR or even a birdie try.