The correct answer is that the North wanted control of the river to divide the South in two, while the South wanted use of the river to keep the Confederacy strong.
The Union had developed, with respect to the Mississippi River, the Anaconda Plan. This plan had as objective to control the adjacent territory to the river in its two margins, in order to split the territory of the Confederation in two.
For its part, the Confederation had as objective to maintain its total control over the river, since by controlling the Mississippi it could establish commercial relations with European countries, like France, that allowed it to cover the expenses of the war, since the river represented a quick exit to the Gulf of Mexico, and therefore to the Atlantic Ocean.