The two chapters do show some stylistic differences. The first chapter is highly structured, uses repetitive motifs, and functions as introduction to the book of Genesis. But stylistic differences need not mean differences in content.
There are several reasons to note the continuity of chapter 2, and the rest of Genesis, to chapter 1. The first and most obvious is that we have received Genesis as a unit. If the author (or compiler) of Genesis wrote Genesis 2 after Genesis 1, which he did, then he knew what was contained in both. It would seem strange that moderns can identify significant differences that he supposedly did not notice.
Secondly, Genesis 1 describes creation repeatedly as good, and after the completion of all of creation as very good. This motif of "good" throughout Genesis 1 is then contrasted with the man's lack of companion which is not good.
Then the Lord God said, “It is not good that the man should be alone; I will make him a helper fit for him.” (Gen 2:18)
To propose a different creation account in Genesis 2 disregards the author's obvious contrast.
Next, Genesis 5 gives the chronology of the antediluvian patriarchs. It begins,
This is the book of the generations of Adam. When God created man, he made him in the likeness of God. Male and female he created them, and he blessed them and named them Man when they were created. When Adam had lived 130 years, he fathered a son in his own likeness, after his image, and named him Seth. (Gen 5:1-3)
This directly quotes Genesis 1
Then God said, “Let us make man in our image, after our likeness. And let them have dominion over the fish of the sea and over the birds of the heavens and over the livestock and over all the earth and over every creeping thing that creeps on the earth.”So God created man in his own image,
in the image of God he created him;
male and female he created them. (Gen 1:26-27)
Thus the command to be fruitful and multiply (Gen 1:28) describes the beginnings of its fulfillment in chapter 5 and the author repeats the earlier text when naming the descendants of Adam. Genesis 5 has obvious continuity with Genesis 2, yet the phraseology also borrows from Genesis 1.
Further, Jesus reads Genesis 1 and 2 together. He does this when refuting the Pharisees on divorce.
Jesus answered, “Have you not read that he who created them from the beginning made them male and female, [Gen 1:27] and said, ‘Therefore a man shall leave his father and his mother and hold fast to his wife, and the two shall become one flesh [Gen 2:24]’?"
The unity of Genesis as a book, the contrast of good with not good, the connection between genealogy and creation, and the unifying approach of Jesus to both chapters, show that it is preferable to read Genesis 2 as an expansion of the Genesis 1 creation account rather than as a different, contrasting creation account.