Your question assumes that Lee was better than Grant, but I argue that the opposite is true. There’s a lot of ways to approach this, but I think the most fundamental one is that Grant learned and adapted, while Lee was stuck in his ways.
I’ll start by telling you a story of a Civil War battle. It was early July, and Lee had been battling Union forces for several days. After some initial Confederate successes, Union forces had withdrawn closer to their lines of supply and took up strong defensive positions on high ground, massing their artillery to pour down on any Confederate assault. Lee decided to mount a frontal assault, which was to be preceded by a massive Confederate artillery bombardment to drive off Union artillery and soften up the defending infantry for the assault. But the attack was a total failure. The Confederate artillery, poorly coordinated and with less effective equipment than their Union counterparts, took heavy losses and inflicted little damage. The infantry assault was also poorly coordinated and forced the Confederates to cross hundreds of yards under withering Union artillery fire, only to then be forced to charge uphill to dislodge the Union defenders. Confederate forces were forced to fall back, suffering heavy casualties and the accomplishing nothing. You might be thinking that I’m referring to the famously disastrous Pickett’s Charge on July 3, 1863, but I’m not. This battle was the lesser-known Battle of Malvern Hill on July 1, 1862.
Exactly a year later, Lee would engage the Union at Gettysburg, yet he made many of the same mistakes. At Malvern Hill, a contributing factor to Lee’s failure was some confusing orders that he wrote to his subordinates which caused them to delay in taking decisive action. Lee made this same mistake at Gettysburg with his order to take Cemetery Hill “if practicable,” unintentionally giving his subordinates leeway to use their own judgement and not assault the hill. His overconfidence in his own artillery's ability to dislodge the Union’s artillery is puzzling, to say the least, given that in both instances, Union artillery was entrenched, possessed a larger total number of guns and far more rifled guns, and had better quality ammunition. And in both battles, Lee displayed an overconfidence that his infantry, through their high morale and fighting spirit, could overcome any volume of withering fire. Unfortunately for many a Confederate soldier, élan does not confer invincibility. If Lee was such a great general, why did he make the same series of mistakes? There’s many contributing reasons, but ultimately it’s because Lee was an inflexible general who couldn’t learn from his mistakes and adapt to changing circumstances.
This is not to say that Lee was never successful, because he was. But much of his success can be attributed to Union failures rather than his own brilliance. Fredericksburg, for example, was a great Confederate victory, but it was enabled by a significant delay in the arrival of the Union’s bridging equipment to cross the Rappahannock River and then by Burnside’s own insistence on a frontal assault. Lee only had one truly great victory, Chancellorsville.
Grant, on the other hand, was a very different general. Both Lee and Grant had fought in the Mexican-American War and were trained in traditional Napoleonic warfare, but while Lee continued to fight this way, Grant quickly changed his style after some early mistakes. He made a name for himself with the capture of Fort Henry and Fort Donelson in February 1862 at low cost, which opened up the strategically important Cumberland and Tennessee Rivers, and his maneuvering to capture Vicksburg in Spring/Summer 1863 to open the Mississippi was brilliant. Unlike Lee, Grant understood that the war would be won through logistics and destroying the enemies’ ability to fight. He coordinated attacks in the east and west to prevent the Confederates from shifting forces between theaters, and he gave Sherman and Sheridan free reign to burn Southern farms and industry to force them to their knees. None of this was taught at West Point or learned in Mexico, Grant learned it during the war.
A fair objection to this reasoning is that Grant had all of the North’s enormous resources to draw on while Lee was force to use whatever the South could scrape up. It’s impossible to say what their decisions would have been were the roles reversed. However, Grant very quickly honed in on how to win the war, Lee did not. Grant understood that winning battles was merely a means to an end, not the end itself. Lee failed to recognize that and so played into the North’s hand. Instead of adapting to fight the North on terms that were in the South’s favor, Lee tried to beat Grant at his own game. To use an analogy, if your high school football team wants to beat the New England Patriots, don’t challenge them to football because you will absolutely lose. Challenge them to baseball instead. Lee insisted on playing football when he should have played baseball.
“Murder at Malvern Hill” by Drew J. Kendall. Military History, August 2002.
The Seven Days’ Battles: The War Begins Anew by Judkin K. Browning
The Artillery Service in the War of the Rebellion by John C. Tidball
The Gettysburg Campaign: A Study in Command by Edwin B. Coddington
The Bloody Crucible of Courage: Fighting Methods and Combat Experience of the Civil War by Brent Nosworthy