You can find the answer to this by reading at the end of Chapter 3 or by looking at Figure 4.1.
Proximate causes are the immediate causes that lead directly to some outcome. So, in Chapter 3, the proximate cause of the Spanish victory was the fact that they had guns and steel weapons and horses and such. But Diamond says that such proximate causes are not the really important thing. What we have to ask is why the Spanish had those things and the Inca did not. That question has to do with ultimate causes. Ultimate causes are the things that make the proximate causes happen.
The difference, then, is that proximate causes are the immediate cause of something while ultimate causes are the deeper causes that allow/cause the proximate causes to happen.