They mean the same thing
"At the end of a long rest" is used in the PHB exactly one time, in the rules for Resting (p. 186) under long rest. All the other references to the time when characters end a long rest use the phrase "finish a long rest" (used sixty times), most often as "when you finish a long rest", or "until you finish a long rest".
One reason is that the rules for resting are not written as addessing the character with "you" directly. The talk in the third person about characters, and say "a character" instead of you, so they cannot use the phrase "when you finish a long rest" without breaking the style of writing.
Both reference the end of a long rest: you finish a long rest at the end of the rest.
D&D 5e is not Magic: the Gathering with exactly defined steps for resolving the order of effects in various phases of a turn and fixed terminology. It's written using natural language, and in the core rules there is not even an explicit rule for resolving simultaneous effects, that was only addressed later in Xanathar's Guide to Everything. There's no need to read more into it, than just editorial laxness.
And, as Thomas points out, interrupting a long rest does not count as ending or finishing a long rest. The resulting beneficial effects only happens when you rested for at least 6 hours out of 8 without major strenous interruptions. Any minor interruptions will not the end or finish the rest, and if there is a major interruption, and you have to start all over again, to finish (or get to the end of) a long rest to these benefits trigger:
If the rest is interrupted by a period o f strenuous activity—at least 1 hour of walking, fighting, casting spells, or similar adventuring activity— the characters must begin the rest again to gain any benefit from it.