Strictly speaking, Alduin IS Akatosh. Akatosh, Alduin, Auri el seem to be aspects of the same underlying entity (Think of the holy trinity in Christian mythology); killing one aspect does not end the whole, indeed it's hard to say if it's even possible to kill any of the aspects. Alduin's soul escapes when his body is destroyed, if he is like any other aedra or daedra, he will undoubtedly come back; he must come back - the prophecy of the world eater remains unfufilled and Elder Scrolls are seldom wrong
Until Skyrim, Alduin was called the Nordic aspect of Akatosh, and some sources claim that Akatosh, Alduin and Auri-el are the same being.
But in the game it's revealed that
Alduin is Akatosh's firstborn. Alduin even calls himself the first son of Akatosh.
Remember that lore is contradicting and contamined by ignorance, superstition and falsehoods. That's one of the good things in TES universe. You cannot trust entirely Nordic beliefs or the Scholars of Tamriel. Their knowledge is limited and partial.