In the photo above, you can see a situation in which a roller is your best friend. The surfaces being painted are large and flat, and it's all one color. The roller will make quick work of those areas.
In the photo below, you see the kind of work that really requires a brush. The area being painted is small and anything but flat. A roller won't paint this evenly, and a brush is a must to get a good finish.
What about a paint sprayer? If you have a really enormous, really flat surface to paint and you're not concerned about how much paint you use, a sprayer can speed up the job. For ordinary exterior painting, the set up and clean up take up as much time as you save by spraying, the sprayer is wasteful, and the finish doesn't look as smooth as the work of a good painter.
So what's the verdict on roller vs. brush? It's easy: you have to have both. Nearly every painting job is a combination of big flat areas and little fiddly bits, even if the fiddly bits are just the places where the wall meets the ceiling.
Use the right tool for the job, and you'll get the best results.