"In the end" is synonymous to finally and is often used without a trailing phrase.
"At the end" however, is usually followed by "of" (if not written, then at least implied). Using it just by itself seems incorrect.
As @Davo correctly pointed out in his comment above, if "of something" is already understood by both parties, then we may skip writing it explicitly after "at the end". That been said, even in this case, we are not using "at the end" just by itself.
[It would be great if someone could elaborate on this more technically]
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