When I am in Texture Paint mode, everything is fine. But when I go into object mode, there are no textures on the wings. Attached .blendhornet.blend (701 KB)
you have a second (empty) material assign to some faces of the wings… remove that second material and all should be fine. 
by the way, you may want to redo your uvmap; just a top projection in this kind of objects is very difficult to paint.
your current brush is also using the same slot texture for painting as the brush texture.