Standard definition TV's display at 720x480 interlaced. Your monitor is much higher than this, and non-interlaced. These conversions tend to reduce the quality of the video on your monitor. Also, ATI cards have always had better TV-Out functions, and typically can match the original quality better than other cards with TV-Out(at least in my experience)
Unless you upgrade to HDTV and all, I doubt you'll get much better quality on your computer. Your problem sounds very typical... I know I've played some of my captured videos back and they looked rather crummy, but then I burnt them to a DVD and the it looked perfect...