Would the Cro-Magnons control South West Asia or not? If they don't then the first area to develop agriculture is in Neanderthal hands. The next three areas to develop in order are China (Neanderthal), the Sahel (Cro-Magnon) and Mexico (probably Cro-Magnon via boats along the coasts of Asia and North America).
If the Neanderthals do so and don't develop agriculture then technological development at a world level will be delayed by about 3500 years. In addition the Sahel agricultural belt is small than the SW Asia belt (which stretchs from the Atlantic to the Pacific.
If they do develop agrciulture when they would be able to gain a technological edge over the Cro-Magnons. The only advantage the latter would have then is that they would be better adapted for hot climates per se than the warm region Neanderthal races.
Hmm... There are Neandertal finds in the Levant, so that might well stay Neandertal territory. I don't think there are any finds in e.g. Pakistan or India, and the agricultural origin sites you missed was the Indus and Nile valleys, so those could be H.Sap even if China and Mesopotamia are H.Neanderthalensis. I don't believe that there are any N. finds from North Africa?