This sounds rather uninformed considering your stated background, but one of the hugest strengths of chess computers is drawing on tons of information about previous games and statistics from memory. Essentially, you'd describe it as having access to books of openings and endgames and other theory so on in the middle of a game.
Why do you think I'm uniformed? Because I haven't mentioned opening books? There is no doubt, that they increase the playing strength significantly. Removing the opening book from a Chess program makes it weaker (say -200 ELO); remove Alpha-Beta pruning and replace it with Minimax, it will hurt terribly (say -500 ELO).
Endgame databases (large collections of solved positions with few stones left) can make a decisive difference in a single game, but be completely irrelevant in others. In 99% of games the average human is lost, even before queens are exchanged. In computer vs. computer, analysis or correspondence play, these databases are more important.
Without knowing your background, I think it makes sense to focus the discussion of Chess AI to areas where it is connected to Civ AI.
Chess computers still cannot compare with humans when this advantage isn't present - eg. Fischer random chess. And players still have beaten top computers recently taking advantage of such flaws - playing to openings the computer doesn't have memorized, in other words.
An expert or world-class player winning 1 game out of a whole bunch by exploiting flaws can't be considered a big success, can it? Show me a player that can consistently achieve a "mere" 33% score in Chess960 against the best software. There appears to be none on this planet.
Chess software demolishes humans, even when not having access to opening books. When playing standard Chess, the human can still rely on his memory of openings, thus usually last a bit longer, before his position collapses.
Have you ever played a game of Chess960 against a strong chess program? The human player can't rely on his memory either and has to think deeply beginning with the initial position.
In 2007 there was an 8 game match,
Rybka vs GM Joel Benjamin, with a human grandmaster playing against the computer with a handicap of 1 pawn. To illustrate, I would compare it with the advantage of starting a Civ game with 1 Worker for free. It is no guarantee that you'll win, but your odds have increased by a lot. The match was won 4,5:3,5 by the computer. Playing without handicap was judged uninteresting, because the winner would have been known in advance.
If Civ was simple enough to be "solved" by an AI (aka able to hold its own against the best human players) it'd be too simple to be worth playing.
Civ can't be "solved" like Checkers. Chess can in theory be solved, but e.g. Poker can't. One reason is, that Civ and Poker are not games of perfect information.