Aperture Priority gives you direct control over the aperture, and as a result the depth of field (DOF).
Fast prime lenses, such as 50mm f/1.4 and 85mm f/1.2 enable you to choose very large apertures for a shallow depth of field. This can help you create those creamy-smooth, out of focus backgrounds that give portraits a professional quality.
Working with such a narrow band of sharpness means that you need to be accurate with focusing - the entire portrait will look soft if you don't focus accurately on the eyes.
