The Confederates would be getting more supplies from abroad; OTL, once Vicksburg fell, there was nothing linking Mexico -> Texas -> Confederate interior, so all their foreign supplies had to come from blockade runners. Also would put them in a better domestic supply situation, being able to get Texan cattle across the Mississippi and to the principle field armies. Serious political pressure on the Lincoln administration as long as the Confederates have the Mississippi, by far the largest artery of international commerce in the U.S., stopped up.
If Lee's 1863 offensive cripples the AotP as badly as in real life or quite possibly worse, he can make sure Chattanooga stays in Confederate hands and Atlanta is properly shielded by sending two divisions west to hold it until winter 63-64. If Chattanooga is in Confederate hands in fall 1864, they have a good chance of winning the war; the ground in East Tennessee is way too hard and rocky to dig proper siege trenches in the winter, and here's tons of high ground for Confederate to fortify. TN sort of draws invaders in two directions; the Mississippi draws them through Memphis to Corinth, while the Cumberland pulls them toward Chattanooga. Inversely, the axes of advance northwards are generally convergent around Nashville.
What matters most of all, though, is Grant IMO. If he suffers a serious defeat, like being separated and destroyed on the west bank of the Tennessee River, and gets exiled to Minnesota to fight Native Americans, then I don't think the Union can win in the Western Theatre. Rosecrans is good, but I don't think it happens without Grant, Sherman, Sheridan, and Thomas at the helm. Henry and Donnelson, Shiloh, Vicksburg, Chattanooga: great prizes each, but Grant lost 35,000 men taking all of them, quite cheap by Civil War standards. Duking it out with Confederates north of Nashville doesn't advance Union objectives one jot, and is sure to run up a much steeper butcher's bill.