The audio interview (where I record my subject using a Marantz or my trusty H4 Zoom) has some big advantages over the video interview. Most people (especially if they're not used to being interviewed) have a deer-in-the-headlights look when you turn on the camera. Not usually the case with your audio equipment.
I've found my audio interviews get me closer to my subject (both literally and figuratively,) and usually get better stories. The entire set up is very intimate. Our knees almost touch. The subject doesn't feel conscious about how they look and, therefore, can be more relaxed. The problem with video interviews, as Mediastorm's Brian Storm says, is "the amount of technology between you and your subject." Without the element of video, it's like you're having a different discussion. Storm and producer Tim McLaughlin discuss some of these advantages in their Audio Interview Basic video.