"At the end" simply means the end time ("They got married at the end of the movie"). "In the end" can mean the same, but it also has the meaning of "ultimately" or "as a result" ("It didn't matter in the end who was right and who was wrong: their marriage was over"; "Company A had more money than Company B, and in the end that was all that mattered").

in the end = finally (adverb) at the end = at the end (propositional phrase)