Compare and contrast Whitman's depiction of the Civil War in his poem "The Artilleryman’s Vision" with the depiction in his letter to his mother.

This is an interesting comparison. Whitman, as far as I know, never directly witnessed a battle in the Civil War; he certainly never was a soldier. But he did serve as a nurse in hospitals for wounded soldiers. His brother George fought in the Union army and saw action in many battles, but Whitman's experience of the war came as a wound-dresser. So it is interesting to read the poem, about a dream vision of a battle, and compare it with his letter (I assume you mean the  letter of January 29, 1865) in which he paints a very different picture of the aftermath of a battle.

The poem consists of two parts: the reality of the artilleryman, safe in bed with the war safely in the past, and the dream of being back in combat. Laying in the dark, he can "hear, just hear, the breath of my infant," marking the beginning of this poem's emphasis on sound. The artilleryman begins to dream of a battle, and most of what he dreams is the sound of the conflict:

"I hear the sounds of the different missiles, the short t-h-t! t-h-t! of the rifle balls, / I see the shells exploding leaving small white clouds, / I hear the great shells shrieking as they pass, / The grape like the hum and whirr of wind through the trees."

The sounds are meant to convey the artilleryman's personal relationship to the action, which nevertheless is epic in scope: 

I see the gaps cut by the enemy’s volleys, (quickly fill’d up, no delay,)
I breathe the suffocating smoke, then the flat clouds hover low concealing all; Now a strange lull for a few seconds, not a shot fired on either side,
Then resumed the chaos louder than ever, with eager calls and orders of officers,
While from some distant part of the field the wind wafts to my ears a shout of applause, (some special success,)
And ever the sound of the cannon far or near, (rousing even in dreams a devilish exultation and all the old mad joy in the depths of my soul,)

The battle is depicted as a great collection of human vitality and emotion, something that arouses the "old mad joy" for being part of something larger than oneself. This is in stark contrast to the peaceful opening of the poem.

It is also in stark contrast to Whitman's own experience of the war, which he writes about to his mother. While in the poem he says "...The falling, dying, I heed not, the wounded dripping and red I heed not, some to the rear are hobbling," in his letter he discusses a particular case of a soldier gravely wounded and stranded on the battlefield for two days. When Whitman asked this soldier about his treatment by the rebels while he was laying on the battlefield, he told the story of a man who helped him:

One middle-aged man, however, who seemed to be moving around the field among the dead and wounded for benevolent purposes, came to him in a way he will never forget. This man treated our soldier kindly, bound up his wounds, cheered him, gave him a couple of biscuits, gave him a drink and water, asked him if he could eat some beef.

Whitman clearly has a lot of experience with "the falling, dying" casualties of the war. This particular extract shows that Whitman had an exquisite sympathy for the wounded and their stories. While there is a kind of nobility in Whitman's description of the kind caretaker, which correlates with his sense of war as a "grand human enterprise," at the same time, this story is meant in part to assuage his mother's fears about his brother, George, and about the brutality of war in general, and about himself. Whereas the "vision" of the artilleryman is a dream of glory, Whitman's letter is a record of the actual reality of war.