Console games are designed to work on TVs, and you can use a monitor, too. However, some special gaming monitors have features that you wouldn't find on a TV, and consoles won't make full use of them.
For example, console games won't fill up a 21:9 ultrawide screen. Ultrawide monitors give me a wider field of view than a TV's traditional 16:9 aspect ratio, which gives me better peripheral vision and lets me spot enemies that I normally wouldn't see with a regular monitor. (Yes, you could say I have an unfair advantage over those who use regular 16:9 monitors.)
Games also look gorgeous and more immersive on an utlrawide monitor, and I can't wait to try Samsung's absurd-yet-tempting super ultrawide 32:9 monitor.
Other monitors offer high refresh rates, like 144 hertz or even 240 hertz, which play games up to 144 or 240 frames-per-second for incredibly smooth gameplay. Console games, on the other hand, are usually capped at about 60 frames-per-second. That's fine for most gamers, but some gamers are sticklers for high frame rates.
And there are some monitors that come with Tobii's eye-tracking technology, which changes your field of view in a game by tracking where your eyes are looking.