Each choice during the 4 parts of the war influences the chiefs'hidden suspicion counter, determining your possible routes in part 5 and a few other things, i have a little chart explaining it quick and exhaustively in my guide ("the war ends - the war (part 5)").
in short, sparing him makes Axel more suspicious (as in "likes you less"), and viceversa for killing him, especially if you met Nauxus, he told you about him, and you still killed him.
Aside from that, there is a quest way later on that becomes slightly longer if he is dead, and with kinda depressing dialogue, but nothing that takes away any reward ("the exotic ingredient hunt").