The statement that "They [amend and emend] have the same definition" is incomplete at best and misleadingly untrue at worse. A look at the entries for the two words in Merriam-Webster's Eleventh Collegiate Dictionary (2003) reveals that amend and emend have one definition in common, but that amend has a number of additional meanings that emend does not:
amend vb {M[iddle] E[nglish], fr[om] A[nglo-]F[rench] amender, modif[ication] of L[atin] emendare, fr[om] e, ex out + menda fault; akin to L[atin] mendax lying, mendicus beggar, and perh[aps] to S[ansk[ri]t mindā physical defect} vt (13[th] c[entury]) 1 : to put right; esp : to make emendations in (as a text) 2 a : to change or modify for the better : IMPROVE {amend the situation} b : to alter esp. in phraseology; esp : to alter formally by modification, deletion, or addition {amend a constitution} ~ vi to reform oneself
...
emend vt {M[iddle] E[nglish] fr[om] L[atin] emendare — more at AMEND} (15[th] c[entury]) : to correct usu. by textual alterations {emended the manuscript}
So, according to Merriam-Webster, both amend and emend can mean to introduce corrections into a text—but amend can also mean (as a transitive verb} in a more general sense to improve, or to alter the sense or provisions of (whether for good or for bad), and (as an intransitive verb) to reform oneself. Emend is not normally used in these last three senses.
Two other reference works confirm the central distinction that MW points out between amend (= "improve") and emend (= "correct"). From Harry Shaw, Dictionary of Problem Words and Expressions (1975):
amend, emend. Amend means "to put right," "to change for the better." We amend by adding or altering, as the noun amendment suggests. Emend, once merely another spelling of amend, has a similar meaning but is properly used only to refer to corrections or changes made in a literary or scholarly work; the corresponding noun is emendation. Both amend and emend are verbs; amend in plural form (amends) becomes a noun meaning "recompense" or "compensation": "He made amends for his careless driving."
And from Adrian Room, The Penguin Dictionary of Confusibles (1979):
amend/emend (change, alter)
To 'amend' something is to improve it. A bill 'amended' in Parliament is thus altered for the better. The very common verb 'mend' is in fact derived from it, with the improving sense still clear in such an expression as 'mending' one's ways. To 'emend' something, on the other hand, is to correct it, remove the errors from it. The word is most often used with reference to a text of some kind that has been corrected. The noun of 'amend' is 'amendment'; of 'emend' it is 'emendation'.
Not to spoil the civics lesson, but a bit of thought about how the real world of politics operates should suffice to convince anyone that amend is often used in connection with making alterations to laws that do not improve them. For example, if the Eighteenth Amendment of the U.S. Constitution—the amendment prohibiting "the manufacture, sale, or transportation of intoxicating liquors" within, into, or out of the United States—improved that document, it's rather difficult to explain how the Twenty-first Amendment—the amendment repealing the Eighteenth Amendment—improved the same document. In the real world, laws are frequently amended into worse forms than they had before; the only accurate meaning amend has under those circumstances is "alter."
One example of the broader reach of amend in the sense of "improve" is its use (in U.S. English) in connection with adding nutrients or soil of contrasting characteristics to existing soil: Gardeners can amend the soil in their garden; they cannot emend it. As the various sources cited above indicate, emend refers primarily to editorial correction of textual errors.
