Both games are great for different reasons.
Six reasons to play Civ 6
1) Ed Beach is the man. I think he's on a creative flourish after dealing with Civ 5's limitations. His detailed-but-fun approach to design does a great job of making mechanics feel thematic, especially in faction design.
2) Successfully takes inspiration from
modern boardgaming design practices that create a more tight but flavorful experience. Makes sense, since Beach himself designed "Here I Stand" and "Virgin Queen".
3) The game spreads its mechanics literally on the map and exposes most of them to be potentially messed up by opponents. It leans into Civ 5's core strengths and expands them into better gameplay.
4) Very clean, stable game even on release day and patches have proven to be consistent and substantial.
5) Retains most of the mechanics from Civ 5 Brave New World while still feeling open to more expansions, which is very exciting.
6) It's a good-looking game with a decent framerate even on older computers. I don't feel a need to upgrade my 2011 PC. Loading time feels slightly faster than Civ 5.
Four reasons to play Civ 4
1) Best game for mods and will likely remain so long past future Civs in the series. Beyond the Sword is amazing for its mod showcase.
2) Good old-school Civ to just play around in for builders and roleplayers while still being a competitive ruleset. It's a great culmination of all that came before it, especially for diplomacy and stacking units.
3) Vibrant visual and musical presentation made Civ 4 the most human game in the series. Leonard Nimoy, Christopher Tin's music selection, "Baba Yetu"*, and the lovely little leaderheads were mindblowing after Civ 3.
4) Very quick game to load up, even in 2005. It's just easier to sneak in a few turns over the lunch hour, especially for PBEM and Pitboss.
*Soren Johnson asked his old roommate Christopher Tin to write an opening theme that would fit the tempo of the sun regularly rising over the Earth in the main menu. Pretty crazy that what was originally a design solution became the first videogame song to win a Grammy and go viral with choirs everywhere.
Here's an excellent talk by Soren about
Civ 4's development process. The mutual respect between art and game design here is great, and informed much of the cool feeling that pervades Civ 4.