In reading about battles in the Napoleonic Wars, it seems that cavalry played a huge role in many battles. Cavalry seems to dominate infantry whenever they catch them in the open and not fully organized. Just the threat of a cavalry charge often cause the infantry to form squares or at least a line of four ranks, which decreases their firepower and make them more vulnerable to enemy infantry and artillery. Heavy cavalry sometimes even charge prepared infantry positions in the middle of the line in an attempt to shock and break them.
In contrast, it seems like cavalry play a peripheral role on the Civil War battlefield. They may clash with enemy cavalry on the outskirts of the battle. Or, maybe they're used for their tactical mobility and fight dismounted like infantry. It seems extremely rare for them to ride down infantry battalions caught in the open or act as shock troops to charge and break the enemy line.
If I'm not making incorrect assumptions, why the difference? The sabers and bayonets are the same. Civil War soldiers have rifles instead of smoothbore muskets--better range and accuracy, but no better rate of fire and neither is likely to get off more than one good, point blank shot at charging cavalry. Is it some other technological change? Differences in training or doctrine? Different, more infantry friendly terrain in the US? Something else?
While it's not one specific reason that cavalry was less efficient, there is a variety of key factors. Terrain, training, experience, technology, and scale in the US Civil War was much different than that of the Napoleonic Wars.
Firstly, the terrain is a bit different depending on where the battles take place. In the South, there are a lot more swampy and wooded areas, or stone dividing walls across fields. This is not to say terrain was more infantry friendly, just less conducive to cavalry.
Technology is a more likely contributing factor to a switch in the role of cavalry, in that the traditional weaponry carried by infantry soldiers in the Civil War was not only more accurate, but had better range. Well-trained soldiers could fire multiple times a minute and their weapons were less likely to jam or overheat. There was a greater prevalence of sidearms as well, increasing the amount of fire put out by the unit in a given amount of time. This made cavalry charges more risky as casualties would increase faster.
Tactics were different as well. Cavalry soldiers were primarily used as scouts, raiders, messengers, and -- if need be -- emergency reinforcements. Many Civil War officers learned from texts written by Europeans about the Napoleonic Wars. While it may seem that this only contributes to the question, there are some other factors. Most US Civil War officers got their start in Mexico. The Mexican-American War had some of the most difficult terrain and consequently made Napoleonic-style cavalry charges impossible. Scrubby, porous land with a high water table, lots of mud and frequent rains, mountain ranges and deserts, even. As a consequence, most cavalry was relegated to an auxiliary role. History also plays another significant part in US cavalry development; the Revolutionary War. It's not as much what the Revolutionary War DID for cavalry as what it didn't. Here again, there was no Napoleonic cavalry, EVEN DURING the Napoleonic Wars. Partially due to inadequate supplies and organization, the rebel Americans didn't utilize direct charges as part of their tactics. Guerrilla warfare ruled.
This abject lack of Napoleonic cavalry has been present in America from the start. Compound this with the common motif of battles starting by surprise and enemies not knowing where the flanks are, and you've got a real issue for cavalry men. Canister shot and better rifles made it more of a risk to charge formations, and the terrain was often obscuring or unstable. Tactics taught at West Point prioritized infantry and artillery, and experience has never proved Napoleonic ways useful. Now bring all of this up against armies that often times numbered in the tens of thousands. Charges just weren't considered profitable even if they had been taught. Units often fought more as a brigade, divison, or corps than as a regiment or company as they did in Napoleonic times. Limits on horses, riders, feed, and saddles kept cavalry units relatively the same size. All these factors combined to define a new role for cavalry in the Civil War, when really -- in America at least -- cavalry and horse troops had never been used in the traditional sense.