There are a number of titles on NES that I can think of such as Empire Strikes Back which only look correct on CRT or anything that does proper NTSC color artifact emulation. (and actually sonic games on genesis too!) I've written a game editor for Apple // graphics which uses NTSC artifacts as part of the editing experience -- and also part of the image dithering/conversion algorithms -- and believe me: It makes a huge difference when you are designing graphics with a 6-color palette where you actually get an extra handful of extra "fringe" colors when using some combinations. If you are still unsure, use an emulator with NTSC emulation (Blargg's is great) and then switch over to plain RGB. There is a huge difference.
Also, a final note on this (Caveat: I am an emulation author and this information is in a very well written wikipedia article on Y'UV if you want to fact check me...) You will NOT EVER get the same colors from RGB than you get from a CRT. The color spaces are different. Emulators can simulate (and in some cases very well) what an analog display does, but it only goes so far. In the NTSC-to-RGB conversion process you wind up having to transform from one color system (Y'UV) to another (RGB) using some rather simple math but then you also have to alias the results to fit the values (which are often outside the 0-255 range). There are colors in the Y'UV spectrum (I'm talking about the Apple colors but there are some on Atari and NES too) that are so saturated that they look completely neon, and those colors actually don't exist in the RGB spectrum at all so you wind up with a rather muted look compared to the real thing.
A scan doubler is okay I suppose for this, but really if you want it to look old school nothing beats the real warm glow of a CRT. If you want to play retro games on an RGB screen, just use an emulator. They're cheaper, and if done correctly you're lucky to ever really notice a difference. :-) I think that you can take a Raspberry Pi and make a dedicated emulator solution for 20% the cost of this scan doubler solution and be just as happy if not happier.