This is two separate topics, so I'll address the War of the Rebellion first.

Outside the Petersburg and to a lesser degree Atlanta campaigns, it's really not that similar at all. Most of the War of the Rebellion was a linear war in Napoleonic style, with the introduction of mass produced rifles enhancing firepower and tactical difficulties faced by the generals without marking any other changes.

The War of the Rebellion has three major elements to it, the Trans-Mississippi Campaigns that get the least attention and saw the greatest use of irregular/guerrilla/partisan war, and some of the forgotten Confederate victories like Mansfield. The decisive Western theater, where the Army of the Tennessee and Army of the Cumberland secured the Mississippi River and ultimately Chattanooga and Georgia, and the Virginia theater where the biggest and bloodiest battles were fought with both sides stalemated until Grant neatly shoved Lee into the trench warfare around Petersburg and strategically nullified him and ultimately did so tactically as well.

Tactically, the new rifles made less difference than one might expect, this is largely due to a blend of factors including the inexperience of most of the US and insurgent officer corps and this showing itself whenever the new formations met other new or veteran formations, and due to a more densely wooded terrain meaning that the full power of linear warfare in European battlefields never quite went off the way that the strategic wisdom of the time would have said anywhere in North America. Rifles, not artillery, were the bulk of combat fatality inflictors in this war, where artillery is the bloodmill of modern warfare from 1905-the present.

The first decisive campaigns of the war in Henry and Donelson and the Peninsular/Seven Days' illustrate the diversity of what it was to be in that war and how different the kind of experiences on the part of armies could actually be. While both McClellan and Grant relied on amphibious means to sustain their offensives, the ships won the fight at Fort Henry and Grant turned an attack the insurgent army launched when he neglected the enemy could do things too into a decisive victory by swift maneuvers and the poor leadership of the insurgent army in the fort.

McClellan's glacial movements worked so long as he faced Joe Johnston who retreated without any intent to fight until told 'fight or your career is over' and then led the botched battle at Fair Oaks and Seven Pines. When he had a general who mirrored his own faults and delusions replaced with a more competent and bloodthirsty and audacious general, Lee proceeded to lose all but one of the Seven Days' tactically but gained a morale advantage over a McClellan who kept deserting his army in the field and read the lopsided victory at Malvern Hill as an apocalyptic threat to his army.

Donelson was a maneuver battle fought with swift advances, retreats, and overpowering attacks by both sides. The Seven Days were a set of nominal maneuvers in Napoleonic fashion intended to place strong forces in McClellan's rear that due to Jackson's corps being present in its absence of military action and in practice a sequence of bloody frontal attacks into the positions of either one corps or the entire Army of the Potomac that as per the usual rule of the offensive meant the attacking army (the smaller Army of Northern Virginia) paid a steeper price in relative terms for its half-victories than the Army of the Potomac did for its half-defeats.

The Army of the Potomac, like the Confederate Army of Tennessee, had difficulties capitalizing on its victories because the influence of McClellan left it prone to treat the Army of Northern Virginia as the force overwhelmingly superior in numbers and firepower, and too politically focused. Hooker did change this some at Chancellorsville, and Meade took the changes further at Gettysburg (which were a month apart).

The gap between this and the Army of the Tennessee, Grant's army, is best illustrated by a battle and a campaign. In the Battle of Bald Hill/Atlanta, the Army of Tennessee encircled it and killed its commanding general on the field at the start of the battle....and the attempt to destroy the Army of the Tennessee ended in its sharpest failure in the entire war,a failure that is hard to explain in most conventional senses as anything but an illustration that the Army of the Tennessee was the best US force of the war at every possible level.

In the Vicksburg Campaign Grant's army won five battles in five days, keeping insurgent forces under Pemberton and Johnston divided, and managed to parlay even the infighting that was unusually high for its high command, in fact at the apex in the war, into strength. It could lead to limitations on bigger victories, as with McClernand at Champion Hill and the start of the Siege, but it did not lead to defeat. The flexibility displayed at Vicksburg does not fit the more plodding images of the Army of the Cumberland and Army of Tennessee, nor the static bloodbaths of Virginia.

Grant's Overland Campaign, to further emphasize the point, was the result not of a deliberate pattern by Grant or Lee, but the failure of a much broader maneuver scheme due to poor leadership that was politically required to be used somewhere but little short of murder for the troops assigned to those commands, and Lee seeking the same decisive victory approach that failed in 1862 and 1863 in much less advantageous situations, leading to his army enfeebled and stuck in the Petersburg trenches.

Ironically the Petersburg Campaign remains the last relatively virgin territory in Civil War historiography and hasn't received as much study as you'd think for the campaign that does the most to foreshadow 1914, in spite of being the longest campaign of the war and pushing the US and insurgent armies in Virginia to the apex of what they would and would not do in modern terms. The Crater, the Deep Bottom battles, and the progressive squeezing of Lee's supply lines, as well as the sustained hammering approach from May 1864 to the surrender of Lee's army does fit into the pattern of modern warfare....but it has not been studied as such due to a variety of reasons, not least that the Southern/Lost Cause view does not like to study where the Confederacy failed. And after the Crater, the litany of failures is very long and very uninterrupted.

The Atlanta Campaign is its own historiographical quagmire, with Hess and Castel and Connelly offering three different and somewhat irreconcilable interpretations of what did and didn't happen there. Connelly's thesis was that Joe Johnston mismanaged a campaign of retreats and was heralded as a defensive genius while doing absolutely nothing to earn this. And that Hood was not quite the idiot he was made out to be but he wasn't a genius by any means, either. Castel in turn argued that Sherman wasn't a genius either and that the campaign as it evolved reflected a mutual limitation of skills between Johnston and Sherman, and that Hood's change to a more decisive approach doesn't validate Johnston...while Sherman' victories said more about Hood than it did about Sherman. Hess's writing tends to be more middle of the road between them.

The USA won the War of the Rebellion in a largely pre-modern sense as the generation of 1914 would have seen it. Only Butler's Army of the James made use of Gatlings in 1864, and it also did innovate with use of wire, but other than that, the USA's way of war fits solidly into the mid-19th Century approaches of Solferino, the Crimea, and 1866. European dismissal of the war was as much from seeing Sebastopol, Solferino, and Koniggratz (to say nothing of Prussia shattering the Second Empire at Sedan) as more of a lesson than the American war.

Sources:

Why the North Won, by Hattaway and Jones

Battle Tactics of the Civil War, by Paddy Griffith

Lincoln's Lieutenants, by Stephen W Sears

Chancellorsville, by Stephen W Sears

Gettysburg, by Stephen W Sears

Grant Invades Tennessee: the 1862 battles for Forts Henry and Donelson by Thomas F. Smith

Six Armies in Tennessee by Steven Woodsworth

Braxton Bragg: The Most Hated man in the Confederacy by Earl J Hess

Army of the Heartland and Autumn of Glory by Thomas L Connelly

Decision in the West: The Atlanta Campaign of 1864 by Albert Castel

Trench Warfare under Grant and Lee: Field Fortifications in the Overland Campaign by Earl J Hess

The Bearss Trilogy on the Petersburg Campaign is some of the newest and most in depth scholarship on it and I recommend it.

World War I is its own animal relative to the War of the Rebelllion. In 1914 all the armies were still largely operating on a scaled up for the new size of the new armies version of the linear warfare that prevailed in the mid-19th Century. In that kind of warfare tightly packed formations relied on arcane drilled maneuvers to move from columns to line of battle to maximize firepower and deployment, but this reflected both limitations of industrial production (or its absence) and of communications technology. Machine guns did not exist in any great scale in the mid-19th century but very much did in WWI.

Indirect fire artillery proved itself a major killer in the Russo-Japanese War, and the lessons of all this and how to adjust to it were missed by all the armies that studied the nature of the land fighting in that war. Artillery in both world wars was the bloodmill of the battlefield,it racked up the highest proportion of combat deaths, and it did so in a fashion that required an adjustment to both sides in the war that was never fully made in the contexts of World War I.

Too, communications technology was primitive relative to WWII to now. Radios existed but were not readily adaptable to major offensives by any army, and this meant a regular pattern where the size and killing power of armies exceeded the ability of commanders to make swift decisions to exploit opportunities as they arose in time for maximum strategic gains. The Russians and Austro-Hungarians had the most obvious defects and the most well known here, but it was not unique to their armies. The German high command could and did suffer these defects to, it certainly did so in 1914 and 1918, and at points in the East. The German Army actually experienced a command collapse in 1914 that was the most epic failure of German arms at any level in either world war, and one obfuscated by wartime and postwar censorship to make the army look less......like it actually was.

It's worth emphasizing, too, that WWI wasn't always a static war. In Salonika, Gallipoli, Flanders, and the Isonzo it was. In the East, Palestine, and Mesopotamia and Africa it was not. The static wars tended to reflect either mutual weakness of command, logistics, and reality-based thinking (Gallipoli and Italy) or overextension and sheer raw firepower (Flanders and France) combined with overstretch and rigid defenses of territory on the part of the Central Powers and the strategic necessity for major offensives for the Allies. It's also worth emphasizing that the Tsarist state made some major errors in deciding where and how to fight its war, in spite of having far more real time experience with modern firepower than any other army. Why that is, and how that is, is a mini-topic in its own right.

That this was more choice than inevitability is neatly confirmed by Operation Alberich, the major strategic withdrawal of 1917 that devastated a wide swathe of France and ensured that the Nivelle Offensive met with a spectacular failure. The choice of Hindenburg and Ludendorff to end the rigid defense of Falkenhayn led to greater flexibility in choices taken but ultimately led to the precise set of decisions culminating in military failure in the 1918 offensives. So too did the emergence of armored warfare from the Somme to the Hundred Days, the arrival and emergence of major US combat forces from the Second Battle of the Marne to the Meuse-Argonne, and the implosion of the German army and its logistical support base behind the firepower-happy columns of advancing troops.

In terms of combat and phases, both wars are completely unrelated to each other, the War of the Rebellion sees a consistent pattern of stalemated bloodshed in Virginia and with the exception of Chickamauga one major insurgent defeat after another in the West, and a largely unbroken string of strategic and tactical defeats in the Trans-Mississippi in the conventional war, with Mansfield and tactically, the first battle in Missouri the major exceptions to an otherwise consistent rule. The cumulative impact of the Western victories collapsed the stalemate in Virginia beyond repair, while the Army of Northern Virginia's over-aggressive over-audacious approach ran into the brick wall of Grant and was denied any chance to keep the initiative, if it was too powerful to be more than shoved into the trench war around Petersburg for most of a year.

WWI was a cumulative attrition war where if one adopted the broader perspective of two warring coalitions the UK and Russia do have a similar pattern in gutting the Ottomans and Austro-Hungarians (the latter far more swiftly than the former) and thus rendering the Germans permanently running themselves to exhaustion to fall ever further behind.....right up until Russia's own issues caused it to go boom and gave Germany a chance to win outright that was more mirage than real and floundered in the abysmal strategic views of General Ludendorff.

However that's a perspective seldom used by the historiography, as there aren't a lot of takes that link in the fuller picture of that war, because after 1917 it's easy to get lost in the two Russian Revolutions and the process ending in the rise of the Soviet Union and the Treaty of Riga and the Ottoman Empire performing surprisingly well against the British Army is hardly something it would want to spread far and wide. The academic focus on the East tends to divorce the Revolutions more totally from the war than is truthfully possible due to both Cold War propaganda and Soviet historiographical influence. Thirty years after the end of the USSR, with Soviet history more broadly integrated into European history, this is starting to change. Key words: starting.

Too, there are continental European (particularly Franco-German) takes on WWI and a British narrative largely codified by Basil Liddell-Hart, whose view of the standard realities of major industrial war in Europe was that the whole thing was a horrible and unimaginably so nightmare that could not be borne, largely from a British perspective of being used to relatively small armies ganging up on colonial wars that lacked even equivalent firepower outside the Boer Wars. Liddell-Hart has had a disproportionate role in shaping the English historiography of WWI, which Holger Herwig is starting to correct, at least somewhat, not least by providing a perspective of the faults of German and Central Powers approaches that Liddell-Hart neatly minimized or outright claimed the Germans and Central Powers were better at waging war than the facts justified.

In contrast, the historiography of the War of the Rebellion is largely shaped, along with the memory, by the Lost Cause and its view of the Army of Northern Virginia, the only Confederate army to be more than a badly led force that got clobbered every time it tried to fight a proper army as opposed to Jayhawkers.

Sources:

The Real War, Basil Liddell-Hart.

The First World War, Germany and Austria-Hungary 1914-8, Holger Herwig

The Marne, Holger Herwig

World War I, Hew Stratchan

The Marble Man, Thomas L. Connelly

The Eastern Front, Norman Stone

All the Tsar's Men: Russia's General Staff and the Fate of the Empire, 1898-1914 by John W Steinberg

A People's Tragedy: The Russian Revolution 1891-1914 by Orlando Figes

The End of Tsarist Russia by Dominic Lieven