Nowadays, Romulus Augustulus is often considered the final Roman emperor, but between the Byzantines, Julius Nepos, and the way the Roman state continued under Odoacer and Theoderic, that's a fairly arbitrary choice. How did we arrive at it? When was Romulus first called 'the last Roman emperor'? Who did the Byzantines consider the final western emperor?
The Byzantines (though obviously they wouldn't have referred to themselves as such) considered Julius Nepos to be the last Western Emperor. This was because he had been recognised as such by the Emperor Zeno in Constantinople.
Romulus Augustulus was a mere child, declared emperor by, and under the total control of his father. His regime never controlled anything more than a small amount of territory around Rome itself. When Odoacer deposed him, young Romulus spent the rest of his life in comfortable retirement, simply because Odoacer recognised just how irrelevant and powerless he was.
After Odoacer deposed him he sent the Imperial regalia of the West to Zeno and said that there was no need for two Emperors. Odoacer declared that from now on the only Emperors would be Zeno and his successors in Constantinople.
Odoacer said that he would rule in Italy as Zeno's deputy but in practice Zeno's control over Odoacer and Italy was effectively nil. So much so that he wound up asking Theodoric to overthrow him and take up the role of ruler of Italy, in the name of the (theoretically reunited) Empire.
Zeno also said to Odoacer that Nepos was still the legitimate Western Emperor but he made no effort to help him regain direct control of any of his putative territory, nor did he send the Western Imperial regalia back to him. When Nepos died in 480, Zeno declared that no successor in the west would be appointed, and henceforth only the man in Constantinople would be the legitimate Roman Emperor.