DVI and HDMI use essentially the exact same format for video. DVI had it first, for use with LCD computer monitors, and was brought over to HDTV with the addition of HDCP copy protection. HDMI came later, and took DVI's video and added digital audio across the same cable. The difference between component and these two is that the DVI and HDMI video is in the digital domain while component is analog (sort of like going from stereo analog to coaxial or optical digital). The benefits there relate to modern video sources and displays: sources with DVI or HDMI outputs typically have their video originating as a digital signal (at least in most cases - the basic channels on your Time Warner box are still analog in some cases), and TV's such as LCD and DLP have digital display engines. Keeping the video in the digital domain all the way through should yield better picture quality.

If your TV has a DVI input, it would probably be worth using the DVI connection where possible. You could test it by feeding the Time Warner box's video straight to the TV using both component and HDMI/DVI to compare the two. You can also consider one of the newer DVD players that will upconvert DVD material to HD resolutions (720p and 1080i). These players are working from a 480i source, so you can't get true HD video, but a player like the Oppo 971H will give you perhaps the best video you'll get out of the DVD format for only $200 so long as your TV will accept DVI or HDMI.

For what it's worth, here's what I'm doing: DVD recorder, universal DVD player (Yamaha S1500), and Time Warner cable box (Pace Micro DC-550) all use component output through the 990, while the Oppo's DVI goes to the 990 and the second DVI input on the 990 is held in reserve for future use (probably a Sony PS3 next year, both for games and to play with Blu-Ray a bit without coughing up for a dedicated player).