According to full preterism, who is 'the dragon' and what are the differences between 'the dragon' and 'the beast'? In some passages from the Bible there exists sentences such as: Revelation 12:9

The great dragon was hurled down—that ancient serpent called the devil, or Satan, who leads the whole world astray. He was hurled to the earth, and his angels with him.

and Revelation 13:4

People worshiped the dragon because he had given authority to the beast, and they also worshiped the beast and asked, “Who is like the beast? Who can wage war against it?&quot;

Since I am assuming that 'Satan', 'the beast', etc are the same entity then
Who is 'the dragon' and what are the differences between 'the dragon' and 'the beast'?
Revelation 12:9 answers the first part of your question. "the great dragon .... that ancient serpent called the devil, or Satan who leads the whole world astray." Most understand that the devil (Satan) also called the deceiver, is in opposition to God.  

The beast is a power. See the beasts in Daniel 7 and 8. This beast obtains it's authority from the dragon. They are not one and the same. Both the beast and the dragon in Revelation 13:4 receive worship. This beast in Revelation 13 comes out of the sea with 7 heads and 10 horns (powers) and 10 crowns (symbolic of authority) on the horns. This beast looks like a leopard with bear's feet and has a lion's mouth. It receives it's power from the dragon for 42 months then one of it's heads receives a deadly wound but this is later healed. There is a second beast in Revelation 13:11 that comes up out of the earth where the first beast had come up out of the sea. It has horns like a lamb but speaks like a dragon. It exercises the same authority as the first beast (authority received from the dragon) and causes all on the earth to worship the first beast that had the deadly wound to it's head which is now healed, and causes all on the earth to receive a mark (of worship). The mark of this second beast is a number, vs. 18, 666. 

There are other beasts that are mentioned in Daniel. If you read Daniel 7 you find 4 different beasts. The 4th beast is a terrible beast with teeth of iron and interestingly, 10 horns. I believe that the beasts in Daniel and the beasts in Revelation help to explain each other. In Daniel 2 Nebuchadnezzar has a dream of an image. It follows by an interpretation of the dream which foretells the progression of the main ruling powers on the earth starting with Nebuchadnezzar and Babylon. There are 5 of them, head - gold, chest &amp; arms - silver, belly - bronze, legs - iron, and feet with 10 toes - of iron and clay. In Daniel 7 you find another dream, this time of beasts and there are four of them, a lion with eagle's wings, a leopard with 4 wings of a bird and four heads. The third beast as a bear on it's side with 3 ribs in it's mouth. The fourth beast is described Daniel 7:7-10. “After this I saw in the night visions, and behold, a fourth beast, dreadful and terrible, exceedingly strong. It had huge iron teeth; it was devouring, breaking in pieces, and trampling the residue with its feet. It was different from all the beasts that were before it, and it had ten horns. 8 I was considering the horns, and there was another horn, a little one, coming up among them, before whom three of the first horns were plucked out by the roots. And there, in this horn, were eyes like the eyes of a man, and a mouth speaking pompous words. Later in Daniel 8 there is another vision of a ram and a goat with the interpretation. This helps to understand the other visions and which kingdoms are to come. The ram with two horns is Medio Persia and the goat is Greece. There is a lot more but this answer is becoming very involved. Blessings on your studies.