According to full preterism, who is 'the dragon' and what are the differences between 'the dragon' and 'the beast'?

The dragon was not the same as the beast of Revelation. Rev. 12:9 identifies the dragon as that serpent of old from the garden in Gen. c.3 - the devil, or the Adversary.

The word “dragon” is drakon in the Greek (Strong's 1404), and means a huge serpent. It only appears in Revelation. In the OT, it is the serpent in Ex. 7:9-10; the representation of Nebuchadnezzar as a dragon in Jer. 51:34; the representation of the idolatrous tribes of Israel in Deu. 32:33; the representation of Babylon in Isa. 27:1; 51:9; and the representation of Pharaoh, king of Egypt in Ez. 29:3.

In the OT, “dragons” and “serpents” were symbolic of rulers of the pagan, heathen nations who worshiped and sacrificed to idols, the unclean things they made with their own hands. Those pagan, idolatrous nations were always the enemies of the Most High.

It is also necessary to understand the symbolic meaning of "earth" and "heaven" in prophetic language, as they do not have the common meaning that is normally understood. In the OT, "earth" was most often used in prophesy for the people of the land of Israel, but could also mean the people of any land to whom the prophet was sent with God's warnings of judgment.

God formed man from the dust of the earth (Gen. 2:7). The prophets spoke to Israel of old calling them “earth.” See Deu. 32:1; Jer. 6:19; Jer. 17:13; Jer. 22:29; Jer. 33:15. Typically, it is best read as "land."

In prophetic language, "heaven" could mean the place where God sits on His throne, but it could also mean the ruling authority and power of a nation. In judgment language it speaks of the governing dominion, palaces and abodes of the world rulers.

In speaking of the judgment against Israel and Judah, Isaiah referred to the light of their heavens becoming dark. (Is. 5:30) Their heavens were their ruling authorities, and the land / kingdom over which they ruled.

Is. 13:1-5 told of God bringing a kingdom against Babylon “from a far country, from the end of heaven…to destroy the whole land.” Here, “the end of heaven” referred to the border ends of the kingdom of Babylon. Babylon’s rule and kingdom was a heaven that was going to be thrown down and destroyed.

Is. 13:10, “the stars of heaven” meant the princes and ruling authorities of Babylon. Is. 13:13 said that God was going to “shake the heavens” and remove the “earth” that is Babylon out of her place.

Is. 14:12 still speaking of the king of Babylon – possibly Nebuchadnezzar, but most probably his grandson, Belshazzar – “fallen from heaven” means he was removed from power and no longer ruled. The name “Lucifer, son of the morning” was referring to the king of Babylon.

Ez. 32:7, in speaking in judgment of the Pharaoh of Egypt:

“And when I shall put thee out, I will cover the heaven, and make the stars thereof dark; I will cover the sun with a cloud, and the moon shall not give her light.” (KJV)

Egypt’s rule and kingdom would be hidden, overturned, and destroyed.

Jer. 4:28, telling the judgment to come upon Jerusalem:

"For this shall the earth mourn, and the heavens above be black; because I have spoken it, I have purposed it, and will not repent, neither will I turn back from it.” (KJV)

The earth was the land, and the people of the land; the heavens were the men who ruled, the Jewish hierarchy, the Sanhedrin Council and priests, their palaces and abodes.

The rulers of a nation lived in higher authority, power, and luxury over the people of the land. Their palaces and their kingdoms were referred to as heavenly places. This is the same reference meant in Eph. 3:10:

“To the intent that now unto the principalities and powers in heavenly places might be known by the church the manifold wisdom of God,” (KJV)

The earthly kings and kingdoms were also called heavenly because it was God who raised them up, allowed them their power and rule. See Ex. 9:16; Judges 2:16, 18; 2 Sam. 23:1; Is. 41:2, 25; Is. 45:13; Jer. 51:11, and many others.

Rev. 12:1 use of “the great wonder in heaven” referred to the Jewish authority and rule of Judah. The woman clothed with the sun was the Israel of old who was pregnant with Christ and His church, born in Jerusalem on the day of Pentecost, with the “moon at her feet” (that is the old Mosaic assembly of Israel), and the twelve stars in her crown (the apostles of the newly born church of Christ)

But, we have to be careful to stay with the context of the scripture to know which heaven is referenced. We see in Rev. 12:7, “there was war in heaven” with Michael and his angels fighting the dragon and his angels… this depicts the war between Rome and Judah… and both rulers / authorities were occupying “heavens” in their sphere of power.

The dragon - devil - was not thrown out of the heaven where God sits on His throne, but was thrown down from a position of power of a ruling nation which was Rome of the first century A.D. Being thrown down to "earth" meant he no longer held the power to deceive the nations.

The dragon in Revelation was a representation of the world power that was the enemy of Christ and His church. It appeared in “heaven”, the sphere of the political world dominion and power of Rome, the empire which ruled over Palestine, the land or “earth” of this prophesy.

There were two beasts in Revelation, the sea beast (Rev. 13:1) and the land beast (Rev. 13:11).

The word “beast” here is “therion” and means a wild beast, a predator that devours the young and the weak. In general, it is a symbol of the pagan world power which was presented in the Roman empire of the first century A.D.

It rose up out of the sea.. that is the gentile nations across the great Mediterranean sea. The seven heads were seven kings / Caesars. The ten horns were the lesser ruling senatorial provinces that did not have the power or rule equal to Caesar. The power to wage war for 42 months was the war against Judea beginning in A.D. 66-67 and culminating in the destruction of Jerusalem in A.D. 70.

Its target, or prey was the newly born, young church of Christ, and His saints.

The blasphemies spoken against God were from the mouth of the pretending Caesars who claimed to be God, and who vowed to destroy all other claimants, and all the seed of David. Specifically, during the great tribulation, the persecution of the saints, it became centered in one man, Nero. (Rev. 13:18)

The symbol of the beast of the sea switches back and forth between the collective pagan empire of Rome, and the representative of that empire, Caesar.

The beast of the earth was the apostate, non-believing Jews who crucified Christ and persecuted His saints, centered in the power of the Sanhedrin and Jewish priesthood.

As in the OT, the word “earth” in Revelation was a metaphor and stood here for the land of Judea, the remnant of Israel. The Jews enlisted the power of the Roman Caesars to both crucify Christ, and persecute the Christians. This is referred to again in Chap. 17 as the woman (Jerusalem) who rode on the head of the Roman beast.

Together, the sea and land beasts were the persecuting powers who killed the newly converted Christians who were a threat to both their power and control of the people.

The Roman Caesars were claiming to be gods, and sons of gods. Their claims were those of impostors, who forced the people to worship them through acts of terror. Those that succumbed to the “mark of the beast” were those that worshiped the Caesars of Rome, and other pagan “gods.”

Satan’s desire was always to deceive the nations / people so that they would turn away from God. This was the war that was prophesied in Gen. 3:15 to be fought between the Seed of the woman (Christ) and the seed of the serpent. This battle waged by Satan was present throughout all of the OT. He desperately wanted to prevent Christ’s birth, and then to destroy Him.

There is much more that is identified in the symbols of the prophesy of Revelation, and it all can be found in the OT. See the posts at my blog Parts 1 - VIII of "The Signs of Revelation" for more on these symbols ShreddingTheVeil.

Also recommended:

"The Great Red Draagon" by Kurt Simmons here

"Reuss on the Number of the Beast" by J. Stuart Russell here

"Symbolism of Prophesy" by J. Stuart Russell here