Immanuel is derived from the Hebrew word "Immanuel" (Isaiah 7:14). "Emmanuel" is derived from the Greek word in the New Testament (Matthew 1:23). Both spellings mean "God with us". Both spellings are correct.
Remember that the Old Testament was written in Hebrew, a language that did not have vowels (e.g. 'e' or 'i'). So when the Hebrew was translated into Greek (i.e. the Septuagint), the translators rendered the Greek "Emmanuel" based on pronunciation not on a particular spelling. Later when the Hebrew vowel were added by Masoretic scholars, a vowel was added which modern scholars typically render as an 'i' rather than an 'e.'
It is important to keep in mind that many ancient languages did not have fixed spellings, but instead spelled words based upon their sound. This was the problem that led Noah Webster to create the dictionary, to establish in America a standard spelling.