Like many people, I like to tall about which games I consider "best", but the truth is that some games (eg. Prismata and Halo) just can't be compared because they fill different roles. In this post I'm going to attempt to come up with a comprehensive taxonomy of these "roles" - the traits that are pivotal but can't be said to be better one way or the other. I believe in three such dichotomies: competitive or casual, replayable or scarce, and turn-based or flowing. I'll discuss the pros and cons of each trait in each category.
These games are played in "matches" or "rounds" that don't build on each other and that you can sit down and play one of whenever you want. They have the obvious draw that, assuming they're designed well, they never get old. You can play these games for a lifetime without ever feeling like you've "finished" them.
Replayable games are usually played against other people, but computer opponents can be created. They can support both competition and cooperation, and have great potential for customizability. Most replayable games have tunable parameters that give the players the freedom to play their own type of match. Dominion, for example, depends heavily on what cards are available, which can be manually picked out by the players or randomly generated.
One significant downside of this type of game is that since you never "finish" a replayable game, you can only experience so many of them. Back in my gaming days being into just Prismata and Go at the same time took up almost all of my gaming time. I don't think I could have gotten into a third replayable game without dropping one of them. And they don't leave a mark on your soul like story games do (see below), so if you do drop one you don't really take anything away from it. All the enjoyment of playing it is in the past.
"Scarce" games are ones that depend on content (such as levels or a story) that must be hand-crafted and which can only really be experienced once. That seems like a huge downside... and it is. But the persistent nature of these games allows levels to build on each other in a way that isn't possible in a replayable game, which can provide a feeling of "progress" that can't exist in those games. These games can also make up for their scarcity by making each level uniquely satisfying in a way that the non-hand-crafted matches of a replayable game can never be. I'll go over two ways of doing this.
This type of game seeks to act as both a work of fiction and a game at once, taking advantage of the special properties of interactive media to enhance the story.
Story games have huge potential for emotional impact. If the story is good, they can leave a mark on the player's soul that leaves them thinking about it for years after they've finished it, or even for the rest of their life. I'm not sure if it's just due to associating the story with another fun experience, but the stories of properly done story games seem to be far and away more powerful than books or movies can.
Although story games are usually single-player, they really don't have to be. The 2003 LOTR Return of the King game shows an example of optional multiplayer in a story game (I didn't use Halo as an example because mutliplayer Halo has some issues with incentivizing degenerate play due to its respawn mechanic). Albeit balancing such a game for both single and multiplayer is usually a challenge, it's far from insurmountable and it's almost certainly worth it for the enormous benefit of being able to play it with friends.
In a well-designed puzzle game each level feels like solving a new problem and learning something you couldn't learn from any other level. It's a lesson that "sticks with you", not as much as a good story, but more than a match of a replayable game.
Another big point for puzzle games is the concept of a level editor. Puzzle games are almost always made with a level editor, and releasing it to the community gives the game more or less the replayability of a replayable game (albeit it's not as efficient because each level still has to be hand-crafted by someone, and community-crafted levels are likely to be diluted with a lot of bad ones). Even if the developers don't release their level editor for community use, the nature of puzzle games allows them to easily crank out new levels once they've built a solid foundation of rules, whereas story games don't become easily extensible like this.
In addition, puzzle games have an advantage in that you can never really lose. Puzzle games have no need to ever send the player back a few minutes like story games sometimes do. They aren't as tilting or salt-inducing as competitive replayable games can be.
As another downside, puzzle games are usually inherently single-player. The mechanics are just incompatible with the idea of multiple players (or it's like Portal 2's co-op mode which was great but required entirely separate levels). And while you can of course give advice, that's not as fun as actually being in the game with someone.
I should also mention that it's possible to mix the two methods above. Portal is an example (albeit the "story" could never stand on its own, while those of most story games could).
All good games involve the engagement of using skills to overcome challenge, but a competitive game tries to be one you can meaningfully improve at and compete with others to be the best. Or, in a scarce game, you can raise the difficulty for a second playthrough to see just how far your skills can go.
A good competitive game can be enjoyed even while you're not actually playing matches of it. You can study it, analyze your mistakes, discuss strategies with other players (even if it's an action game there's usually some element that can discussed like this), and, at least in a replayable game, train apprentices. Replayable competitive games are also often great for spectating, which allows you to enjoy the game without the stress or commitment of actually playing.
Competitive games almost inherently have a higher peak of fun than casual games. Both can be fun to play, but only one can provide the incredible euphoria of winning a match against a top player or beating a boss that killed you a dozen times.
As a direct corollary of that, though, competitive games inevitably can be upsetting. They can lead to a lot of salt and/or tilt over one's losses. Good game design can mitigate these (by making the game feel as fair as possible), but it can't completely remove them.
The only other bad thing about replayable competitive games is that you can't really dabble in them, because if you know you don't plan on getting "into" it, you can't appreciate the depth in the way that you have to to fulfill the point of such a game.
(I've been changing my philosophy on "casual" games lately. There's sort of a double meaning in the way I use the term. The category I'm talking about here is not the one I bash so heavily in that article.)
Casual games focus on accessibility and are meant for playing with groups. The objective they're aiming for over competitive games is that you don't need to be "into" a casual game to play and enjoy it. You can introduce someone to this game and have an enjoyable match with them right from the start, even if you're an experienced player. The main ways they achieve this include: a ruleset simple enough to be learned while playing, team play or even fully cooperative play (eg. Codenames: Duet), no player elimination; and, if not fully cooperative, making it hard to tell who's winning until the end and some sort of way to give a weaker player a chance against a stronger player.
That last one can is usually achieved through randomness. But Prismata, although it's a competitive game, shows an alternative: performance variance. Each game of Prismata has a different set of units available for purchase and therefore a different optimal strategy, and since your "skill" is essentially how well you play in each set averaged out, odds are not bad in any single match that a weaker player will pick a better strategy than a stronger player.
For examples of good casual games, I'd list Codenames and Dominion. I used to consider Dominion a competitive game, but not anymore.
The downside - and it's a huge one - is that they can't really be taken seriously. You can't get "into" a casual game, because it doesn't have the depth of a competitive game (and usually also not the fairness either).
Turn-based games have a few obvious advantages. They can be interrupted when they need to be, and the turn timers can be adjusted to suit individual preferences or casual play. The biggest downside of turn-based games is the bore of waiting for your opponent in PVP, which gets worse the more players you have; like with the upset of competitive games, good game design can mitigate the problem but can't completely remove it.
The other big downside of turn-based games is that they have difficulty avoiding skill ceilings - in a turn-based game, if you know the right move, that's all there is to it, no execution involved; and that limits how good you can get at it. The game's enjoyment is reduced when there are no nontrivial decisions to make for a few turns, even if the player hasn't completely solved the game. Relatedly, turn-based games almost inevitably suffer from unwinnable situations; this isn't a huge problem but is less satisfying than having it be always possible to come back with excellent play.
It is unfortunately impossible to prevent coaching in an online turn-based game (whereas that can't really be abused in a realtime game).
Realtime games have an enormous advantage in the elimination of waiting time, nonconsensual coaching, and often the elimination of skill ceilings and unwinnable situations too. They can also more easily be more exhilarating since decisions are made more rapidly, and they don't tend to suffer from skill ceilings or unwinnable situations, because execution is so variable that there's almost always room to out-execute even a very skilled opponent. In Dragon Ball FighterZ, I've seen matches where a player reduced to a smidge of health on their last character proceeded to wipe their opponent's team of three high-health characters left. That game doesn't have a negative feedback loop.
But realtime games lose the time management skill, introduce potential problems with lag or network latency, and the biggest issue is how it can look like the game is bullshitting you (such as fighting game situations where you could swear you were blocking but somehow theenemy hit you with a blockable attack anyway) even when it's not and you don't have time to look at it and see how it's fair.
One thing designers can do to get around the problem of "illusory bullshit" is to allow the player to watch a replay of their battle. This is a good idea, but doesn't completely fix the problem: it takes time to watch it, the player won't always care enough to spend that time, and they might still feel it's bullshit if the keys you're telling them they pressed don't align with their memories. Even if their memories are wrong, even if the game actually is compeletely fair, it's the player's experience that counts.
Realtime strategy games don't have the illusory bullshit problem so much; but a lot of people find them dissatisfying because they dilute the strategy with other skill factors, making it about something other than finding out what the best strategy is (which itself has a lot of emotional appeal). RTS games also lose the advantage of removing skill ceilings and unwinnable situations.