Let me see if I can explain this. First off, if you play a fast enough game or run into a situation in a slower game that might require tighter timing or reaction, it can become pretty apparent that there's a significant difference. Casual players don't notice it as much because they're not as attuned to it... this can actually be a small burden/handicap for those who can (I hate versing my friends at Smash. I'm the best but LAG).

A(n) HDTV is best if you want the best look for what you're playing. Combine this with the correct third party accessories for N64 and you can end up with some incredibly crisp HD N64 display, but this comes at the cost of visual delay. HDTVs are designed to take a video input, process them behind the scenes to make them look as nice as they can, then output it (Via LEDs or similar) to the display. Unfortunately, this process takes time. Not long (ranging from 0.05 to 0.3 seconds-ish in my experience), but in some circumstances, this can really mess things up. Furthermore, if you go from one HDTV to another, chances are they'll have different delay times. If you build up muscle memory for one display, it could potentially get messed up and/or undone when you use another. I wouldn't recommend this for Speedgames or competitive games, but it makes for great cinematic experiences in the right room.

Cathode Ray Tubes (CRT) are the best if you want gameplay to be as accurate and lagless as possible. It takes the video input, and spits it out (via photons) to the screen. There's no processing done, no need to remember any pixels, and working with Photons directly helps a lot too. They can be a pain to configure or transport, and they don't typically look very pretty, but I struggle to notice any visual delay when I press a button (no doubt there is some, but it's minute). Using the same muscle memory analogy from the previous paragraph, nothing will be lost going from one CRT to another, making gameplay more consistent. I find that the best benefit that comes from this is that reacting to something in game becomes easier than trying to anticipate something... just remember to lift from your knees.

Anyone who reads this, good on you. Also, correct me on anything I get wrong please. tl:dr IMO, CRTs are more consistent and better for competitive and speed games, HDTVs are prettier and are better for movies.