(1) It is the same thing I point out. I have highlighted the Greek in blue. Again the difference is only one letter.
ἐν ἀνθρώποις εὐδοκία
ἐν ἀνθρώποις εὐδοκίας
That is the same thing I posted above. Here it is again.
The TR has: ἐν ἀνθρώποις εὐδοκία.
The CT has: ἐν ἀνθρώποις εὐδοκίας.
Note the sigma on the end. That bigger difference you of which you speak is because of only one letter.
The Net Bible also points out one is in the genitive case and the other is in the nominative case (highlighted in green). The two phrase are exactly alike except for that one letter and that accounts for the change in translation.
(2) The Net Bible cites A Textual Commentary on the Greek New Testament, Second Edition by Bruce M. Metzger. Metzger makes the same points.
"The difference between the AV, “Glory to God in the highest, and on earth peace, good will toward men,”
and the RSV, “Glory to God in the highest, and on earth peace among men with whom he is pleased!”
is not merely a matter of exegesis of the meaning of the Greek, but is first of all one of text criticism. Does the Angelic Hymn close with εὐδοκία or εὐδοκίας?
The genitive case, which is the more difficult reading, is supported by the oldest representatives of the Alexandrian and the Western groups of witnesses. The rise of the nominative reading can be explained either as an amelioration of the sense or as a palaeographical oversight (at the end of a line εὐδοκίας would differ from εὐδοκία only by the presence of the smallest possible lunar sigma, little more than a point, for which it might have been taken – thus ⲉⲩⲇⲟⲕⲓⲁⲥ)."
(3) In Mounce's grammar (3rd ed) at the beginning of each chapter there is a section called Exegetical Insight. At the start of chapter seven this text (i.e. Luke 2:14) is examined. It states:
"'Peace on earth, good will toward men' (Luke 2:14, KJV). You have probably all received Christmas cards containing this part of the angels’ song to the shepherds on the fields of Bethlehem. But most modern translations read differently: 'on earth peace to men on whom his [God’s] favor rests' (NIV); 'and on earth peace among those whom he [God] favors' (NRSV). The difference between the KJV and the others is the difference between the nominative and the genitive.
The Greek manuscripts used to translate the KJV contain εὐδοκία (nominative), whereas the older manuscripts used to translate the modern versions contain εὐδοκίας (genitive) —literally translated, 'of good will' or 'characterized by [God’s] good pleasure.' In other words, the peace that the angels sang that belonged to the earth as a result of the birth of Christ is not a generic, worldwide peace for all humankind, but a peace limited to those who obtain favor with God by believing in his Son Jesus (see Romans 5:1). What a difference a single letter can make in the meaning of the text!"