In essence, a solid stain is a paint. They are both films that sit on the surface. A solid stain is very different than a penetrating stain, which actually absorbs into the wood and doesn't sit on the surface.
Solid stains and paints both require that the wood be primed. Both could be subject to peeling and both are opaque films that completely cover whatever you are painting with a uniform coating. However, while they are essentially the same, solid stain is different than paint in a couple ways.
First, solid stains and paints have different thicknesses. Stains are thinner than paints. The benefit of a thinner film is that it better highlights the texture of the wood, while the benefit of a thicker film is that it bridges cracks and imperfections to form a smoother surface.
Second, solid stains are typically very flat. The advantage of a very flat finish is that is can look good with a rough surface like striated shakes. The downside of the flat is that it is less durable. And while exterior paints often come in a flat finish, the recommended finish for the house body is typically a low lustre. The low lustre finish has dull sheen that wears extremely well in the weather but may not look the best on a rough surface.
So while paint and solid stain is essentially the same type of coating, there are differences that lead to different uses. Solid stains are most often used when the wood is rough, or when you want to accentuate the wood texture. And paint is used when the surface is smooth or you want the longest lasting coating to protect your wood.
Another way to think about the difference is this: since the solid stain is thinner, it doesn't last as long, but it is also easier to redo when it is time. And since paint is thicker, it lasts longer, but is more work to redo when it is time. On average, a solid stain lasts about 7 years and a high quality exterior paint will last about 25 years.