Everything said so far makes sense. My anime DVDs looked far better on my old EDTV plasma than on my computer screen, because the resolution matched DVD resolution perfectly. But here's a story...
I just did the whole media PC/HD upgrade, because I wanted a 50" and to watch 1080p anime on it and play games from my recliner. I got a decent new PC, AMD PHENOM II X2 550 AM3 with an ATI HD 4870/1GB card (HDMI output). Plasma is still to my eyes richer in color, so I got last year's best Panny TH-50PZ850U. Out of stores now, but nice deals online, and it's apparently the very last consumer model that gives you access to the pro settings.
Anyways... this was after buying, used, a 2-yr old top of the line Mits 50" LCD, which was Editor's Choice in 2007. I re-sold it within 3 days. Just not a plasma, really. I can't give up my phosphors and viewing angle. Plus the detail in blacks was gone compared to my old 42" Panny plasma display, already 5 years old. I only had a Pioneer Elite DVD with upscaling at the time, and DVDs were literally unwatchable whether the Mits TV or the player was handling upscaling. I mean large pools of pixelation, that followed Batman around as he walked.
Enter the new Panny, the media PC with Blu-ray drive and the ATI HD card, and a new Pioneer VSX-1018AH-K. Let the card do the upscaling via GPU, so 1080p goes straight to the receiver. Even anime DVDs (SD), which obviously were unwatchable on the other setup, are crisp. Very occasionally, I get minor pixelation on lines, but almost not noticeable at about 7 feet, my usual viewing distance. I also have a 24" Dell monitor that goes beyond 1080p. Occasionally, I've watched a bit of anime on it, and it looks good, but definitely not comparable to my new setup. I've yet to see an LCD that achieves the richness of plasma (YMMV), and now with 1080p and proper upscaling, there's just no comparison.
Moral of the story, you can get the CRT-TV/DVD-anime experience, but you must have the upscaling quality to get the experience you want at HD resolution levels. My setup (PC, TV, Receiver) cost me about $3,500 total, which I consider a steal. Add $500 for a decent speaker/subwoofer set, and $200 for misc. crap, and you're set for another 4 years. I don't need a separate Blu-ray player, and I have full access to online services and streaming (even CR "fake 720p" streams look fine). Not to mention it rocks for playing games as well. With 50" 1080p, I get to play Bioshock in my kind of style

You don't need to spend what I did though. Just make sure that somewhere in the chain is high quality upscaling for old DVDs and low-resolution video files, and again I suggest plasma because it uses phosphors, so is closer to the CRT experience. And I would suggest upscaling at the earliest possible point rather than relying on the TV or receiver to do it (e.g, PS3 for DVDs).