I heard an interesting analogy that shed some light on the situation for me, so I'll share it here. Majorana fermions are topologically based; let's look at what topology sort of "means".
Topology looks at the bigger picture. If you have a balloon, no matter how much you blow it up, take air out, or tie it up in knots (if you're a balloon artist), it still doesn't have holes in it. To have holes would make it fundamentally different. You can stretch and shrink and twist a sphere all you want, but it's never going to turn into a donut. If you take a donut, though, you can twist that into all sorts of things with holes - but you can never make something without holes, like a sphere, or with two or more holes.
Another example of topology looking at the bigger picture. Take a balloon (again) and zoom in on its surface. Even though the balloon is curved when you zoom out, when you're zoomed in, it looks like a 2-d Euclidean plane. If you zoom in on a circle, it looks like a 1-d Euclidean plane. The little twists and turns don't matter in topology.
Let's bring this back towards Majorana fermions. Let's picture a system where we're registering if the electron goes all the way around a tree or not. It doesn't matter whether the electron has a really squiggly nutty path or a just a simple circular path - it still goes around.
The noise introduced in these systems might make the electron's path squiggly or it might not, but it doesn't actually matter. It still goes around. That's where the advantage in Majorana fermions lies - the noise doesn't affect it.
Obviously this isn't rigorous; I'll try to add more that sheds light on that as I have time.