It's really quite simple, especially if you use PC video card and monitor resolutions as an analogy.
Yes, all standard DVDs are 100% digital, but at nowhere near the resolution of HD. They have a fixed resolution of 720x480 pixels, which was a good match for even high-end analog television sets. Consider that the basic equivalent of an old VGA-equipped PC. Or, using a digital camera analogy, DVD would be the equivalent of a 0.35 megapixel camera.
HD, on the other hand, can have up to 1920x1080 pixels of resolution. By way of comparison with standard DVD, that would be the equivalent of today's state of the art video cards for the PC, or a 2 megapixel digital camera (six times the resolution of standard DVD).
Both HD DVD and Blu-ray are designed to store movies on DVD-sized discs at full 1920x1080 resolution (I won't go into the differences between 1080p, or progressive, versus 1080i, or interlaced, since that is a whole other discussion, and it doesn't have any real bearing to this topic). The key here, though, is that neither HD DVD nor Blu-ray have any advantage over each other in terms of resolution or overall picture quality. They are basically identical.
To be able to store that much data, both HD DVD and Blu-ray use more advanced technology to increase a disc's storage capacity. A standard single-sided, dual-layer DVD has a maximum capacity of just over 9 GB. For HD DVD, that capacity has been increased to 30 GB, and for Blu-ray, 50 GB. They can also use newer, more efficient compression technology to store their movies, making the effective capacity of each disc even greater.
Both HD DVD (I have one) and Blu-ray do look noticably sharper and more "film-like" than DVD, but I won't go so far as to say that it will be as striking a difference for most people than it was when we moved from VHS to DVD. That is something you'll have to judge for yourself. As far as I'm concerned, though, there is no comparison.