Why are there different divisions of the Ten Commandments?

Good question, I didn't know for the different divisions. Searching about it I found the following information very helpful.

That there were “ten words” is expressly stated (Ex 34:28; Dt 4:13; 10:4); but just how to delimit them one from another is a task which has not been found easy. For a full discussion of the various theories, see Dillmann, Exodus, 201-5, to whom we are indebted for much that is here set forth.

A. Josephus is the first witness for the division now common among Protestants (except Lutherans), namely,

a. foreign gods,

b. images,

c. name of God,

d. Sabbath,

e. parents,

f. murder,

g. adultery,

h. theft,

i. false witness,

j. coveting.

Before him, Philo made the same arrangement, except that he followed the Septuagint in putting adultery before murder. This mode of counting was current with many of the church Fathers, and is now in use in the Greek Catholic church and with most Protestants.

B. Augustine combined foreign gods and images (Ex 20:2-6) into one commandment and following the order of Dt 5:21 (Heb 18) made the 9th commandment a prohibition of the coveting of a neighbor’s wife, while the 10th prohibits the coveting of his house and other property. Roman Catholics and Lutherans accept Augustine’s mode of reckoning, except that they follow the order in Ex 20:17, so that the 9th commandment forbids the coveting of a neighbor’s house, while the 10th includes his wife and all other property.

C A third mode of counting is that adopted by the Jews in the early Christian centuries, which became universal among them in the Middle Ages and so down to the present time. According to this scheme, the opening statement in Ex 20:2 is the “first word,” Ex 20:3-6 the second (combining foreign gods with images), while the following eight commandments are as in the common Protestant arrangement.

The division of the prohibition of coveting into two commandments is fatal to the Augustinian scheme; and the reckoning of the initial statement in Ex 20:2 as one of the “ten words” seems equally fatal to the modern Jewish method of counting. The prohibition of images, which is introduced by the solemn formula, “Thou shalt not,” is surely a different “word” from the command to worship no god other than Yahweh. Moreover, if nine of the “ten words” are commandments, it would seem reasonable to make the remaining “word” a commandment, if this can be done without violence to the subjectmatter. See Eerdmaus, The Expositor, July, 1909, 21 ff.

^{Orr, James, M.A., D.D.: Orr, James (Hrsg.): The International Standard Bible Encyclopedia : 1915 Edition. Albany, OR : Ages Software, 1999}

The reason why different groups view the Ten Commandments differently is because each group assumes different priorities from the text. Some of the commandments are long because they include a justification or explanation, so it's natural to shorten those to a single phrase when compiling the list. It is these expanded commandments that leave room for different number orders.

A lot of Protestants say that Catholics lumped "Thou shalt not make unto thyself graven images" in with the first commandment in order to avoid the issue of venerating icons. This shouldn't be the case, since Jews have put "Thou shalt have no other Gods before me" with "graven images" since antiquity. During the time of Exodus, the Jewish worldview at the time stands out among it's Middle East peers. The first commandment in their version lays the basis for monotheism. It is the foundation of all the commandments.

Christians see the Jewish first command as a preface. For a long time, Jewish understanding of the Scriptures was lost to the Christian world, whether by lack of contact or simple ignoring. Therefore, in Protestant and Catholic minds, there is no need to establish monotheism, and the division of ten can only be in the rest of the list. At the same time, there is a different wording near the end of the list in the Exodus/Deuteronomy versions: Catholic tradition emphasizes the latter, where coveting your neighbor's wife comes before coveting your neighbor's house, so from a Protestant perspective, which emphasises the earlier decalogue, it doesn't even enter the mind to split them, so they go to the most natural section. Furthermore, Protestants are protesting, among other things, the veneration of icons, so they emphasize the commandment against graven images.

Even if making carved idols is a sub-unit of the first commandment, then it still would seem apparent that the rest of the explanation and justification for that commandment is concerned primarily with making idols. For ancient Jewry this is understandable since false worship always included idols in the world around them. Catholics also know that they only worship one God, so they don't see the reason to separate two ideas that are so connected. Protestants are critical of Catholic practices after the Reformation period, and they don't have the option of making two different covets because of the way the text they use is worded. Protestants feel like this commandment of God (even as a sub-unit of one of the ten, it still is a command) has been understated, and they elevate its status.

The divisions are competing merely because of historical reasons. They don't actually contradict each other. The Law is summed up in "Love the LORD your God" and "Love your neighbor.".