It is alleged that there are three different versions of the Ten Commandments, two in Exodus and one in Deuteronomy. Let's take care of these starting with Exodus 34, which is the easy one.
And the LORD said unto Moses, Hew thee two tables of stone like unto the first: and I will write upon these tables the words that were in the first tables, which thou brakest...and Moses rose up early in the morning, and went up unto mount Sinai, as the LORD had commanded him, and took in his hand the two tables of stone. And the LORD descended in the cloud, and stood with him there, and proclaimed the name of the LORD....And he said, Behold, I make a covenant: before all thy people I will do marvels, such as have not been done in all the earth, nor in any nation: and all the people among which thou art shall see the work of the LORD: for it is a terrible thing that I will do with thee. Observe thou that which I command thee this day: behold, I drive out before thee the Amorite, and the Canaanite, and the Hittite, and the Perizzite, and the Hivite, and the Jebusite. Take heed to thyself, lest thou make a covenant with the inhabitants of the land whither thou goest, lest it be for a snare in the midst of thee: But ye shall destroy their altars, break their images, and cut down their groves. For thou shalt worship no other god: for the LORD, whose name is Jealous, is a jealous God: Lest thou make a covenant with the inhabitants of the land, and they go a whoring after their gods, and do sacrifice unto their gods, and one call thee, and thou eat of his sacrifice; And thou take of their daughters unto thy sons, and their daughters go a whoring after their gods, and make thy sons go a whoring after their gods. Thou shalt make thee no molten gods. The feast of unleavened bread shalt thou keep. Seven days thou shalt eat unleavened bread, as I commanded thee, in the time of the month Abib: for in the month Abib thou camest out from Egypt. All that openeth the matrix is mine; and every firstling among thy cattle, whether ox or sheep, that is male. But the firstling of an ass thou shalt redeem with a lamb: and if thou redeem him not, then shalt thou break his neck. All the firstborn of thy sons thou shalt redeem. And none shall appear before me empty. Six days thou shalt work, but on the seventh day thou shalt rest: in earing time and in harvest thou shalt rest. And thou shalt observe the feast of weeks, of the firstfruits of wheat harvest, and the feast of ingathering at the year's end. Thrice in the year shall all your menchildren appear before the Lord GOD, the God of Israel. For I will cast out the nations before thee, and enlarge thy borders: neither shall any man desire thy land, when thou shalt go up to appear before the LORD thy God thrice in the year. Thou shalt not offer the blood of my sacrifice with leaven; neither shall the sacrifice of the feast of the passover be left unto the morning. The first of the firstfruits of thy land thou shalt bring unto the house of the LORD thy God. Thou shalt not seethe a kid in his mother's milk. And the LORD said unto Moses, Write thou these words: for after the tenor of these words I have made a covenant with thee and with Israel. And he was there with the LORD forty days and forty nights; he did neither eat bread, nor drink water. And he wrote upon the tables the words of the covenant, the ten commandments.
It is usually argued that here, God said He would make a set of tables containing the words that were on the first set; yet the ten commands that follow do not match the usual set of "Thou shalt not..." commands.
The answer: Ex. 34 contains a ritual Decalogue that is likely meant to be parallel to the ethical Decalogue, and Ex. 34 as a whole is a narrative of the renewal of the covenant following the golden calf incident.
The ritual Decalogue is an addition in light of that incident, and note that Moses is commanded to write them, whereas it is God who will write the new copy of the ethical set (34:1, 28; the subject of "he wrote upon the tables" in the last verse is God -- it is because these critics understand the subject as Moses that they are missing the point).
With that, what about the differences between Exod. 20 and Deut. 5? Let's run them in parallel and see what the issue is:
Thou shalt have no other gods before me. -- Same words in Deut. 5.
Thou shalt not make unto thee any graven image, or any likeness of any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth: Thou shalt not bow down thyself to them, nor serve them: for I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me; And showing mercy unto thousands of them that love me, and keep my commandments. -- Same as in Deut. 5, other than perhaps some difference in vocabulary.
Thou shalt not take the name of the LORD thy God in vain; for the LORD will not hold him guiltless that taketh his name in vain. -- Again, the same.
Remember the sabbath day, to keep it holy. Six days shalt thou labour, and do all thy work: But the seventh day is the sabbath of the LORD thy God: in it thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates: For in six days the LORD made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the LORD blessed the sabbath day, and hallowed it.
This is where the objections begin, and it's not about minor differences in vocabulary (for example, the ox and ass are specified, versus the generic "cattle"); rather, the charges emerge in the difference in reason for the Sabbath. In Exodus 20:11ff, it is said, it is becase of creation's seventh day; but in Deuteronomy 5:15 we are told:
And remember that thou wast a servant in the land of Egypt, and that the LORD thy God brought thee out thence through a mighty hand and by a stretched out arm: therefore the LORD thy God commanded thee to keep the sabbath day.
Critics would do well to recall here, though, that Exod. 20 and Deut. 5 are addressed to different sets of people with different experiences -- the first set had just been released from Egypt; the second set had never known slavery but were born during the Exodus; moreover, the Deuteronomy covenant would rule those who came afterwards who never knew the toil in Egypt.
The initial generation had been saved from Egyptian slavery by way of incredible miracles; for these people, the role of God as Creator was a central truth -- and they would need no reminder that they didn't get a day off in Egypt. But now, as the Israelites are about to cross the Jordan and enter into the unknown perils of taking over the Promised Land, they are better reminded of the Sabbath in light of the former (and unknown by experience) plight as slaves.
Of course, the original release marked the redemption of the Israelites, but mainly their creation as a covenanted nation. Note also that there is a difference in the word used: Ex. 20 says to remember (zakar) the Sabbath day, and tells why God blessed the Sabbath; Deut. 5 says to keep (shamar -- the meaning is to protect or guard) the Sabbath, but rememeber (zakar) the slavery in Egypt.)
Honour thy father and thy mother: that thy days may be long upon the land which the LORD thy God giveth thee. Thou shalt not kill. Thou shalt not commit adultery. Thou shalt not steal. Thou shalt not bear false witness against thy neighbour. -- Only minor differences in vocabulary here in the first of the set.
Thou shalt not covet thy neighbour's house, thou shalt not covet thy neighbour's wife, nor his manservant, nor his maidservant, nor his ox, nor his ass, nor any thing that is thy neighbour's. -- Here, the order of "wife" and "house" is different, and "field" is added in Deuteronomy -- no surprise, since the Exodus group was nomadic; the next generation would settle down into agriculture.
And that's it. There's just no reason to make an issue of the variations.