Well what type of cavalry? Napoleonic era cavalry varied hugely.. And they were almost like chess pieces in that all types had a different use on the battlefield.

The main types most commonly seen on the Napoleonic battlefield were: Cuirassiers, Heavy Dragoons, Light Dragoons, Lancers and Hussars.

Cuirassiers - Heavy Cavalry with thick breastplates riding large warhorses but which moved slowly and tired quickly. Armed with a heavy chopping saber. Think the equivalent to tanks.

Heavy Dragoons - Heavy or Medium Cavalry depending on the unit. Not really Dragoons by the Napoleonic period in that they weren't armed with carbines in general just sabers. More lightly armoured and on slightly lighter horses than the Curassier.

Light Dragoons - Medium to light cavalry depending on the unit. Mounted Infantrymen who as standard dismounted to fight. Not armoured and only really effective on horseback when chasing down broken opponents. The closest parallel to US cav in the civil war. They would in the US cav where available be riding quarter horses but many regiments just ride anything they could get hold of as the North had a major shortage of horses throughout the war.

Lancers - Light and very fast cavalry equipped with light sabers and reasonably well able to charge other cavalry light cavalry.

Hussars - Light very fast cavalry on expensive horses for the most part by the Napoleonic period. Designed to fight fleeing infantry hacking down and inflicting horrendous wounds with curved sword and arguably the most feared type among infanteers.

There were other sorts of militia cavalry too such as Yeomenry but they don't need a mention.

---

If the Civil War, and the US Cavalry happen to have repeaters it's probably no contest. Because it's bringing a machine gun to a musket fight.

In Napoleonic terms US cavalry have always been light Dragoons. Unarmoured and not really designed to fight cavalry on horseback. They generally fought as mounted infantry that would stop, dismount and fire.

Fighting on horseback the US civil war cavalryman wasn't awfully effective simply because it just wasn't what they were for. If they dismounted it then becomes us civil war infantry vs Napoleonic cav and so falls outside your question.

The average US civil war cavalryman was also largely much less highly trained than a Napoleonic cavalryman. They were often hastily raised units and the generals of the time only regarded them as useful either for duties like raids on supply depots or scouting which was their primary role.

Does that help answer your question?

Oh and actually charging a square was generally futile. Square was formed specifically to repel cavalry but was vulnerable to artillery so in an ideal world you'd put your cavalry half a mile away from an infantry regiment which was then stuck in square and then have the square sufficiently disrupted by artillery that it no longer was coherent in its formation or had holes, at that point you'd charge with heavy cavalry...

Cavalry charges generally were impossible in the civil war since the accuracy, rate of fire and range of rifles with mineé balls meant you simply could not charge across open ground (the only ground you can effectively charge on) with a cavalry regiment against an infantry regiment and expect anything but the first line of cavalry to be unhorsed or shot while still a couple of hundred yards away and the regiments second line then would become entangled messily and be sitting ducks as the Rifles reloaded and fired another volley.

If you ambushed infantry or outnumbered them several times a charge might be plausible but the US at the time was heavily wooded and with lots of obstacles like rail fences (unlike the open fields where most Napoleonic battles happened) meant charging just simply wasn't practicable most of the time.

Sources Haythornthwaite, Philip. Napoleonic Heavy Cavalry & Dragoon Tactics. Vol. 188. Osprey Publishing, 2013.

Haythornthwaite, Philip. Napoleonic Light Cavalry Tactics. Osprey Publishing, 2013.

Griffith, Paddy. Battle tactics of the civil war. Yale University Press, 1989.

1.No and yes.

"No," in the sense that some cavalry units in both wars fulfilled some of the same basic roles. A French Hussar in 1815 and a Confederate cavalryman in 1862 would both have spent time patrolling, scouting, and screening the main movements of their army. Cavalry (particularly irregular) also acted as raiding forces. A trooper in Mosby's cavalry and an Russian cossack in 1812 would have been responsible for raiding enemy supply lines and harassing rear areas.

"Yes," in the sense that there were very few Waterloo-style cavalry charges during the Civil War and very few major cavalry-on-cavalry fights that were part of a larger engagement between two armies (the cavalry fight at Gettysburg is an exception). For the most part, ACW cavalry fights were distinct cavalry-on-cavalry battles, with little or no infantry present. The Yellow Tavern or Brandy Station battles are good examples of this. There are a handful of cases where cavalry did charge infantry as part of a large battle. Jeb Stuart's charge at 1st Bull Run, stands out as an example. But there's never anything that resembles Ney's massive cavalry charge at Waterloo. In short, cavalry generally didn't play much of a role as a mounted battlefield force.

Which brings me to one of the bigger features of ACW cavalry tactics. Cavalry commanders like John Buford or Nathan Bedford Forrest tended to use their cavalry as a sort of mobile infantry. In line with Forrest's dicta to "get tahr fustest with the mostest," ACW cavalry commanders often used their cavalry to outmaneuver enemy infantry. Once in a good position, the horsemen could dismount and fight on foot. Since many cavalry troopers carried fast-firing Sharps or Spencer carbines, dismounted cavalry could often outshoot infantry! And, if necessary, the troopers could remount and fight somewhere else.

Another major feature of ACW cavalry operations was the use of cavalry in massive raids. Kilpatrick's and Stuart's used of thousands of cavalrymen in massive raids on enemy supply lines and rear areas stands out as a major feature of ACW cavalry warfare. To my knowledge, at least, such practices were rarer in Napoleonic times.

2.Why these differences? Why were there so few massed cavalry charges? I'd say weapons, terrain, training, and livestock all played a role.

Technologically, the minie ball, and the rifle, musket, shell-firing artillery made massed cavalry charges against prepared enemy lines prohibitively expensive. Consider the failure of Russian cavalry to even [i]reach[/i] the "Thin Red Line" or the failed charge of the Light Brigade at Balaclava in 1854. Now, that's not so say that these made cavalry-on-infantry/artillery engagements impossible, rather that it made them more difficult. And obviously, these changes didn't dramatically effect cavalry-on-cavalry fights, which largely remained sabre-on-sabre deals until the late 19th century.

Terrain-wise, the eastern United States is heavily wooded in many places. Any open areas were often divided by stone walls and fences. Massed cavalry need lots of clear space to maneuver and many Civil War battlefields simply didn't have it.

Training probably played a role as well. It took decades for Europe to build up an officer class, doctrine, equipment, and institutional knowledge that enabled cavalry to be maneuvered and fought en masse. ACW armies simply didn't have a comparable depth of institutional knowledge, at least at least at the start of the war.

Finally, there was a shortage of specialized cavalry horses. Napoleonic forces could use whatever horseflesh was at hand. But if you wanted really dominant cavalry, you needed good, specialized horses. The light horses used by Hussars or the massive chargers used by Cuirassiers required special breeding, selection, and breaking. The United States in 1861 had great breeding stock for farm horses and carriage horses, less so for real warhorses.