Why participate in an organized century ride when I can ride the route on my own? I have never ridden a century ride before. This June, a friend and I plan to ride our first: The Montauk Century Ride. But I'm wondering, why should we commit to paying $100+ to ride this route in relatively heavy bike traffic, from a starting point not chosen by us, on a day that could be rainy for all we know? Why not ride the route on a day chosen by us? It seems to me riding on our own might provide a more convenient experience and more meaningful challenge.

What am I missing here?
Details depend on the ride, but usually organized rides provide things that you'd have to provide yourself. This one appears to provide a lot. This is covered by the catchall term "support", as in a supported ride.

The support that everyone enjoys is the food and drink along the way ("The Best Rest Stops") - scroll down on the page you linked, and you'll see that in this case it's a lot better than just a powerbar and a refill for your bottle. It looks like there's even food, beer, showers, and massages after the ride!

Support generally also includes some kind of assistance in case of breakdowns. Presumably you can fix flats, but say your tire is too damaged, or some component on your bike breaks. At the very least, you can get a ride in a support vehicle - and they might be able to help you get back on the road in some cases too. There's some about the S.A.G. (support and gear) service on their "ride day" page. They can also just give you a ride if a century turns out to be too much for you to handle.

Of course, another part of the appeal of rides like this is the "relatively heavy bike traffic". Some people like riding with plenty of cyclists on the road, and perhaps to hang around with after. And you don't have to worry about getting lost, or being visible to drivers.

So the supported ride will be a very different experience. Think of it as the challenge of riding a century without any details to worry about, and a bit of a party along the way and at the end. If you know you can handle the distance self-supported, don't care about riding with others, and it's more convenient to ride on your own, go for it. But if you go for the supported ride, your money isn't just disappearing into thin air.