From reading the two wikipedia articles, they had different opinions on how the air force should be used in war theaters.
Douhet offers (emphasis mine):
Douhet believed in the morale effects of bombing. Air power could break a people's will by destroying a country's "vital centers". Armies became superfluous because aircraft could overfly them and attack these centers of the government, military and industry with impunity, a principle later called "The bomber will always get through". Targeting was central to this strategy and he believed that air commanders would prove themselves by their choice of targets. These would vary from situation to situation, but Douhet identified the five basic target types as: industry, transport infrastructure, communications, government and "the will of the people".
In contrast, Wever offers:
Wever outlined five key points to air strategy:
- To destroy the enemy air force by bombing its bases and aircraft factories, and defeating enemy air forces attacking German targets.
- To prevent the movement of large enemy ground forces to the decisive areas by destroying railways and roads, particularly bridges and tunnels**, which are indispensable for the movement and supply of forces
- To support the operations of the army formations, independent of railways, i.e, armored forces and motorised forces, by impeding the enemy advance and participating directly in ground operations.
- To support naval operations by attacking naval bases, protecting Germany's naval bases and participating directly in naval battles
- To paralyze the enemy armed forces by stopping production in the armaments factories.
So, in essence, where Douhet envisioned that the Air Force was the only thing that mattered, with wars ending quickly without much ground or naval operations by basically bombing the civilian population to the point where they revolt and sue for peace, Wever viewed the Air Force primarily as a means to support ground and naval troops.
Or put another way the first was a proponent of systematic carpet bombing as a strategy in its own right (and was proven wrong in WW2), while the second was more of a proponent of precision bombing of strategic targets.