Applying sealcoating with a sprayer is the preferred method of professionals. It's fast, provides good coverage and does a superior job of penetrating the surface. A special sprayer is required, however, and it can get messy, especially in tight quarters.
Applying sealcoating with a roller is an attractive option because it requires no expensive equipment. You can simply pour some out on the prepared pavement and roll it on like paint. This job does heavy-duty rollers, however, as a regular paint roller cover won't last long applying such a thick material to a rough surface.
Unless you have a huge area to cover, applying sealcoating with a sprayer is probably best left to a professional. For a driveway or patio, rolling should work well. Be sure to use a quality product and good rollers, and take your time. You might also consider using a brush or squeegee like the pros do for edging and detail work.