Both are English forms of the same Hebrew name, written with four letters: YHWH (יהוה). Yahweh is how many scholars reconstruct the original pronunciation. Jehovah is the form English Bibles have used since William Tyndale in 1530, including the King James Version.
The Divine Name King James Bible uses Jehovah to stay with the King James text.
Where Jehovah comes from
Ancient Hebrew was written without vowels. When scribes later added vowel marks to the text, they gave the four letters of the name the vowels of Adonai, a reminder to read "Lord" aloud. Reading those letters and vowels together gave the Latin form Iehouah, and from there the English Jehovah. Tyndale used it in his translation of the first five books of the Bible in 1530, and the King James translators kept it in four verses.
Where Yahweh comes from
Yahweh is a scholarly reconstruction. It draws on how some early Christian writers spelled the name in Greek, and on the short form Jah. Many modern translations and reference works use it.
Which is right?
Nobody knows the original pronunciation for certain. Most Bible names come into English in a form that differs from the Hebrew: Jesus from Yeshua, Jeremiah from Yirmeyahu, Isaiah from Yeshayahu. Jehovah has been the English form of God's name for nearly 500 years, and it's the form readers of the King James Version already know from Exodus 6:3 and Psalm 83:18.
Why this edition uses Jehovah
The point of the Divine Name King James Bible is to restore the name without rewriting the King James text. Using the translators' own form keeps the book consistent with itself, from Genesis to Malachi.
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