Granted Civil War history isn't my main field, but the answer isn't as simple as "Sherman had the idea and Grant resisted." And likewise the reverse isn't completely true.

Grant had learned the value, and practicality, of moving through the Southern countryside during the Vicksburg campaign. He also began to understand at that time that the Southern popular will would need to be defeated along with the field armies. But he also understood that Lee's ANV was a physical representation of that will, and that the Army of the Potomac would not likely close with and destroy that army without his physical presence. Actually, both Sherman and Grant were wary of the possibility of moving an entire army through the deep south, but over time both began to see the possibilities of that movement. Grant seems to have grasped the need to carry the war to the deep south first, but Sherman certainly came around quick enough. He was the one who was wary during the Vicksburg campaign of cutting loose from the supply lines and living off the county, but had clearly changed his tune when the Army of Tennessee cut loose through the deep south.

Both Sherman and Grant (and by extension Sheridan and many other senior Union leaders from the Western Theater) grasped the need to shatter Southern popular will and support for the cause. Such a realization seems to have come with slower speed and clarity to those Union commanders who spent most of their time fighting in Virginia.

Sherman's campaign certainly changed to a tune of punishment when they left Georgia and entered South Carolina.