I don't believe that Chess will ever be superceded by any single Chess variation, not even the 3D Chess popular in Star Trek, but I do envision a future in which Chess variations in general will rival Chess in popularity. For many years, Chess was the only Chess-like game I could play on my computer, and that helped cement its popularity with me. I have since discovered programs that will play other Chess variations, and now Zillions of Games, which will play nearly any Chess variation, including those I make up, like Cavalier Chess here. I still have a strong interest in Chess, and I don't regret buying Chessmaster 6000 shortly before I heard of Zillions of Games. I am interested in many variations of Chess, and I like having lots of variations better than being tied down to just one variation. So I don't expect Cavalier Chess to replace Chess in popularity, nor do I think it should. What I do expect is that it will become a popular Chess variant, sharing a place among other popular Chess variants, mainly because it is a good game.
The most important difference between Chess and Cavalier Chess is the power of the pieces. In Cavalier Chess, the pieces are a lot more powerful, and each player attacks or controls over half the squares on the board from the beginning of play, including some of the squares occupied by enemy pieces. In Chess, it takes time to build up to a middle game where strong tactical moves are possible. In Cavalier Chess, a Nightrider could capture the enemy Queen on its third move, without the Queen ever moving. This can be stopped with proper play, but it illustrates the sort of possibilities that exist in Cavalier Chess from the beginning of the game. Chess is divided into an opening game, a middle game, and an endgame. These are all present in Cavalier Chess, but in Cavalier Chess, middle game play acquires greater dominance. The increased power of the pieces allows for stronger tactical maneuvers, and it also makes it harder to safely move pieces across the board, keeping play as balanced as it is in Chess.
The endgame in Cavalier Chess is more touch and go than the endgame in Chess. Because of the greater mobility of Cavaliers and the Knight King, a passed Cavalier is almost a foreign concept in Cavalier Chess. Cavaliers can more easily maneuver to stop each other from promoting. Since what a Cavalier promotes to depends on which square it lands on, the first player to get a Cavalier promoted has less assurance of a win than the first player to get a Pawn promoted in Chess would have. For example, I played an endgame which was down to the Knight Kings and an even number of Cavaliers on each side. I promoted first and got a Nightrider, but my Nightrider couldn't stop an enemy Cavalier from eventually promoting to a Marshall, and I lost the game.