"but as soon as I forget about the fact that I am or am not watching an HD source and just go ahead and watch the content, I very quickly forget I'm watching DVD."
The difference is detail. DVD or DVD-quality rips on my 47" 1080p television I sit 2 metres from are lower quality than HD content. In movies, things are "fuzzy" around edges (or, worse, jaggy if I get aliasing effects) -- while HD movies have complete detail (I could pause the playback and count the change on a table, or read the flavour text that's in a lot of movies on props, etc). With non-HD, you're only getting a vague sense of the set and setting around the actors. Corpse Bride is so clean in 1080p, that you can see details that remind you they're little dolls again (like dust sometimes showing up in a shot, which DVD would just blur away). Older HD content (eg: Taxi Driver) shows more film grain, and reminds me more of home cinema experience than DVD does.
In gaming wise, aliasing is a lot more obvious. A 1080p image from the Xbox 360 looks a lot nicer and more detailed than the Xbox Originals, or something like a PS2 game.
In both cases, the difference is pretty obvious.