"And there are seven kings: five are fallen (Egypt, Assyria, Babylon, Medo-Persia, Greece), and one is (Rome), and the other is not yet come (the Beast/Antichrist of Rev. 13:1-10); and when he cometh, he must continue a short space. And the beast that was (Rev. 13:1-2), and is not (Rev. 13:3), even he is the eighth (Rev. 13:4-10; the Antichrist, like Judas Iscariot, is called "the Son of Perdition," when he is indwelt by Satan - Jn. 17:12; 2 Thes. 2:3), and is of the seven, and goeth into perdition" (Rev. 17:10-11).
Why the difference? Perhaps like Matthew and Luke, who differed with the names listed, and the starting point of Jesus' genealogy (Mt. 1:2 begins with Abraham, while Lk. 3:38 begins with Adam), Daniel, wanting to flatter Nebuchadnezzar, began with him, while John, being in exile, was free to speak of both the past and the future. Regardless, at least four of the empires are included in both prophecies. And while obviously not one of the eight empires, they both also speak of Christ's Millennial Kingdom (Dan. 2:44-45; Rev. 20:1-7).
Something that has always confused me in the past, is that Daniel's text speaks of the Greek Empire being depicted as the "his belly and his thighs" (v. 32), and the Roman Empire represented by "His legs of iron, his feet part of iron and part of clay" (v. 33). So which is it? The "thighs" are said to be a part of the Greek Empire, and yet, if the "thighs" are not considered a part of the legs, then the Roman Empire began with two separate, disconnected parts. Historically, Rome was a single entity, and did not split until A.D. 1054.
So, I used my handy-dandy Young's Analytical Concordance to look up the word "thigh," and here is what I discovered: the Hebrew יַרְכָּה (yarekah), used in Daniel 2:32, appears only once in the singular form in all of Scripture. Because it has as a root, (yarek), which is translated "thigh" 21 of the 28 times it occurs in the Bible, the translators decided "thighs" was correct. Ironically, Daniel did not use the "dual form" of the noun, also the Hebrew yarekah, which would make much more sense, if he intended the image to depict two separate entities! As it turns out, Daniel may have purposely or "accidentally" described the Roman Empire as starting out together at the "loins," and then, alter dividing into two. Yarek is actually translated "loins" in Genesis 46:26 and Exodus 1:5.
Moral: God's Word, like an onion, should be continually "peeled!" - 2 Tim. 2:15