For 8 and 16-bit era console games (nes, genesis) it's a matter of preference. The graphics were designed with many different monitors in mind, so display isn't an issue, and emulation is so good at this point, especially on the 8-bit side of things, that it's very hard to tell the difference from the real deal
For 32/64-bit consoles (ps1, n64) playing it on actual hardware is usually much better than emulating. Most modern PCs aren't really powerful enough to be able to emulate 3D graphics hardware entirely in software, and as a result most of them aren't really emulators as much as they are interpreters, converting the game's system calls to native ones in real time (eg. running an n64 game in openGL). You're not emulating these games as much as you are using a program that lets you run them on your own system, and as a result accuracy is obviously an issue.
Things are flipped on the home computer side of things. Most games late-70s/early-80s micros relied on some form of hardware trickery that simply can't reasonably be reproduced via emulation. The apple ][ used NTSC color artifacting to display color, as did most games for early IBM PC compatibles, and the 6581 SID used a hardware fault to play sampled audio. None of these things work quite right in emulators because they relied on a lot of non-digital hardware to work.
Later home computers like the Amiga and post-VGA PC compatibles, meanwhile, emulate like a dream since a) most hardware faults were ironed out, and the hardware itself was powerful enough that games didn't need to use hacks to get good graphics and sound anyway, and b) the bigger range in hardware being written to (a game written for the A1000 would be expected to run of an A500), meaning that emulators can afford to be looser when it comes to accuracy and still play pretty damn identically to the original system
Games written for computers with x86 processors like IBM PC clones, the NEC PC-98 line, and systems like the Tandy 1000 that were compatible with but distinct from the IBM PC are usually better emulated, primarily because most of the hardware on modern machines is compatible with them, and all an emulator needs to do is smooth out the differences
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