The resolution (total number of pixels) of each frame in Standard Definition video is a maximum of 720x540 pixels. This is dependent on the format you shoot, so for instance mini-DV will be 720x486 pixels per frame. DV-CAM will shoot the same pixel resolution, but each frame will have less compression applied.
Newer still digicams have much higher resolution than that. For instance a Canon PowerShot G9 has a max resolution of 4000x3000 pixels.
So while it's possible to pull a still off video and make a "picture" that will look decent in small size on say a web page, it takes quite a bit of "magic" to make a print of that file look even half decent in say a 4"x6" format. And no matter what "magic" you apply, such a print will never have the level of detail and clarity that a high-pixel-count still shot will.
So I guess it's a question of what kind of compromise you're willing to live with
Of course, the video function of newer digital still cameras is supposedly getting better - I don't have much personal experience with that though. The G9 for instance will shoot 1024x768 pixel video, but only at 15 frames per second. For full motion, 30 frames per second video, the best a G9 will do is 640x480 pixel video. I don't know if the housing for these digicams will allow you to switch between still and video while you're under water, but *if* it's possible, then you would be able to shoot high res stills as well as video in a quality that you may find acceptable.
Hope this helps.
Henrik
(edited for clarity and accuracy)