To amend is (1) to change for the better, (2) to put right, or (3) to alter by adding. The word’s corresponding noun is amendment.
Emend means to improve by editing (especially a text). Its corresponding noun is emendation. Emend is rare because it’s mainly confined to contexts related to professional writing and editing.
The two words share a root in the Latin ēmendāre, which means, roughly, to remove fault. The older amend came to English, around the 13th century, via French, where the e in the Latin word had become an a several centuries before. Emend, which came to English a couple of centuries later, is more directly derived from the Latin source, so the more Latin spelling is intact.
Amend, incidentally, is the source of the verb mend, which today is similarly defined but more broad and informal than its source verb.
El-Beshry said the committee was appointed to amend the constitution in a way that would help maintain the country’s sovereignty and promote democracy. [Sify]
The bill would amend the state motorcyle helmet law to say that only people under 21 would be required to wear a helmet. [KEZI TV]
The Norton Anthology editors emend the text to contain a comma after “merchants” rather than a colon. [Walter Pater via Project Gutenberg]
Second, all critics have agreed to condemn the digression in which Theobald advertised his ability to emend Greek texts. [Hugh G. Dick via Project Gutenberg]