Whenever someone tells you that the only time you'll notice the difference between HD DVD/Blu-ray and standard DVDs is if you have two TVs side by side hasn't seen these comparisons.
As you mouse over the screencaps, you can see that colors are brighter, details are increased, and a lot of the fuzziness is gone. And that's with the HD images downsampled. Feel free to use this as a guide to convince your significant other to let you go HD.
Fellowship of the Ring - HD vs DVD [Cornbread]