I think the only real difference is psychological.
A Nation may start in conquering of various peoples, in federation, or any other method... but its dominant culture thinks of the entire area as One Thing... one economic unit, one political unit, ultimately, one people.
e.g. the US started as a bunch of various states, but the Union mentality was that the various states were ONE THING above and beyond that.
Nations States strive to convince everyone within them that they have a common identity.
I think Empire differs in that the dominant culture sees itself as separate and superior to its subject cultures.
Britain ruled over India and other cultures, but it never saw those cultures as British... but as lesser cultures that needed, required, or were better under British control.
In this sense, the USA treated European immigrants and its subject states as a nation/state... but it treated Native Americans, the Philippines and other holdings as an Empire.
Those in the position of citizens of a Nation/state are seen to be served by their State.
Those in the Position of subject to Empire, are seen to serve the Empire.
Thus there was a difference between being British, and being part of the British Empire.
The Irish and the Scots have never felt they were fully included in the identification as "british".
To the British, Britain was their "Nation". It was THEIR Empire. It served them.
Everyone else subject to British rule felt they served the empire.