You absolutely want to use a brush for most exterior surfaces, and definitely the siding and trim. The only time to ever use a roller is on a large flat surface (think drywall, or plywood paneling) -- in the image you provided I might use a roller for the white soffit, but that's it and only if it's a large house.
Aside from avoiding unsightly "orange peel" effect that a roller will undoubtedly case, applying the paint back and forth with a paintbrush will work the paint into the surface of the product in a way that a pass over the top with a roller will not. This is especially true with wood surfaces, but generally true with most surfaces. Just imagine being paint on a relatively smooth surface. Wouldn't it be easier to peel if you were rolled on in one large sheet, than if you were brushed on and worked into the surface with each brushstroke?
Exceptions to the rule, for exterior surfaces are:
- Concrete - I roll foundations and cement block with 1" nap roller
- "Cedar Shake" siding (think Cape Cod) where I apply the paint with a 1-1.25" nap roller, then backbrush to work it into the siding and catch drips (this is much faster than brushing, and higher quality than spraying)
- Large soffits or areas where I'm working over my head for long periods
- T111 siding, vertical barn siding, etc, but in all cases I backbrush
Definitely buy a 4" brush for this project.