Internet chess is a different experience than table chess. I conjecture that it will with time influence both its development and the rules of chess. Table chess is typically slow. It is a much more psychological and instinctual experience. You can see the terror in the opponent's face when you sacrifice the bishop with check. Chess is by nature very strategical and, in the opening phase, very theoretical, spiced with tactical themes. It demands deep analysis, a sincere effort, both during the game and before, not to become boorish wood-chopping.
However, Internet chess is essentially rapid chess, yet without the psychological atmosphere of table chess. What's more, the very valuable ingredient, which is the hard toil to produce a memorable game, is lacking, too. This means that the experience of chess is devaluated and the game is experienced as somewhat monotonous. Sooner or later it grows slightly dull. It is as if chess isn't really designed for Internet play, while it isn't a superficial game. You must dig deep into it to really appreciate the game. You need good play from both parties, and a true effort, in order for it to be really engaging.
I have also played Chinese Chess (Xiangqi) on Internet servers. Xiangqi is really cut out for the Internet. It is, by nature, a fast and very intuitive game. It is all about tactics, while strategical planning plays no part at all. Compared with chess, it is superficial. Yet it is entertaining as tactical intricacies start immediately and it goes on until either party is mated. You never get a breather, because it is all about grabbing the initiative and to attack, before your opponent does the same. It is fast-paced, since you never have to find out how to solve difficult problems by devising a deep plan. Instead, you solve problems by devising short tactical lines involving some finesse. Sometimes the tactical situation is so impermeable that you cannot possibly calculate it, even if you have loads of time. At these occasions one must resort to intuition, which draws on experience.
Xiangqi opening theory is much different from chess theory. In Xiangqi, it's like all the variations are thrown into the same bucket. It's more or less the same themes, which means that you cannot benefit much from studying theory. There aren't really different lines that differ radically in opening strategy. Some are more aggressive and some are slower, that's all. In chess, if a player decides to change opening from, say, the Stonewall to the Benoni, it's like learning a different game. If he has played hundreds of Internet games in the Stonewall, he is becoming slightly bored, and that's why he has decided to change opening. The "Stonewall universe" of variations is quite interesting, provided that players make an effort of strategical planning. But this seldom happens in Internet play. So he decides to change to the Benoni and finds that he has regressed to amateur level. This is yet another nail in the coffin for his chess passion.
This never happens in Xiangqi. In opening play anything goes. Provided that the move isn't immediately refutable, it is wholly playable. As Xiangqi isn't partitioned into different "opening universes", like the Benoni and the Stonewall, the whole field of tactical themes are always present, which is always the same as all other openings. So there are no constraints and no monotony, which often occurs in Internet chess, especially when you have played the French Exchange for the umpteenth time. There is no such thing as a boring or a drawish opening in Xiangqi.
As Xiangqi opening theory isn't essential, cheating is not a big issue. Xiangqi softwares are weaker than human players. They seem to pose no threat at all to Grandmasters. I don't know why it is so. I don't think the game is more complex. It could have to do with the ramification of variations, i.e. how the search tree looks, as it is not partitioned into smaller "opening universes". More importantly, there is always a tactical trap beyond the search horizon, since tactical traps are always present.
A feature of Internet chess is that some games are boring while some are engaging, it's up and down all the time. Games are engaging when they are well-played, especially when they are theoretically correct, and no player blunders. However, many games degenerate into dreary wood-chopping that is essentially meaningless as there is a lack of intellectual content, only the monotony of realizing a material advantage. This never happens in Xiangqi. A loss of a pawn in Xiangqi doesn't mean anything, and even if a player blunders a whole piece the game often continues, as there is no monotonous way of realizing a material advantage, but the player must resolve the problems by tactical measures. Sometimes you can blunder a whole rook. The tactical complexity is such that the opponent can miss out on one of the many tactical traps, and then you're back in the game. The conclusion is that the game is fun even if there has earlier been very bad play. The search after tactical finesses continues. There is no such thing as a boring game of Xiangqi. Monotony doesn't exist. However, the game can be accused of being superficial. Chess is much deeper.
I fear that Xiangqi will continue to expand on the Internet at the expense of chess, if Internet chess cannot transform and adapt. We must begin to differ between Internet chess and table chess. These two arenas endow chess with very different qualities. A suitable variant for Internet rapid chess should be developed; a variant, perhaps, that has some of the characteristics of Xiangqi. For instance, a Xiangqi pawn cannot be blocked. It might be enough to change the rule for the pawn, so that it cannot be blocked so easily, what contributes to the slow and strategical character of chess, so unsuitable for Internet rapid chess. Such a variant will not make Fide-chess obsolete. As a belligerent complement to drawish and strategical Fide-chess it will contribute to the popularity of chess. I have myself developed a few proposals on my chess homepage.