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The main point Fuller makes is that Grant was the first general to understand the totality of warfare in the modern age, including the role of political expectations. He also was a superior strategist and campaigner to Lee, although Lee was probably the better battlefield tactitian. Lee had the advantage in the Overland campaign of fighting on the defensive, and Grant was aware of the approaching elections and the need to produce a result, rather than the traditional Army of the Potomac stalemate, or worse. He additionally had responsibilities for overseeing the Western and Valley battlefronts.
The main point to remember when considering the careers of the two men is that, if my memory is correct, of the three armies that surrendered during the War, Grant received two of them.
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Grant was by no means an incompetant general, but his main attribute was tenacity; he won through stubborness more times than anything else. He nearly let the garrison of Ft. Donaldson get away (and would have if anyone had listened to Forrest), he came within a hair's breadth of losing his entire army at Shiloh, he was frustrated time and again at Vicksburg by an inferior general, his plan to defeat Bragg at Chattanooga was convoluted and shouldn't have worked, and Lee out-generaled him on multiple occasions. What made Grant superior to any other Union general (save perhaps Thomas) was his refusal to give up. Grant understood the advantages he worked with and knew how to use them to win. Put him on equal footing with Lee and the story may well have been different.