There are a number of minor differences which can now be explained from this perspective, although, if we did not have the background I have already provided, these small differences might seem more ambiguous. Matthew identifies the apostle John as “his [James’s] brother” (Matthew 17:2). Mark omits this identification. Matthew makes this same identification of John in Matthew 4:21 and 10:2. So he added the identification in the Transfiguration narrative.
The second time Moses and Elijah are mentioned (Mark 9:5 and Matthew 17:4), Moses is mentioned first and Elijah second in both Mark and Matthew. In Mark, the first time Moses and Elijah are mentioned, Elijah is mentioned first (Mark 9:4). We may assume Matthew reversed this order in the parallel passage (the second mention of Moses and Elijah; Matthew 17:3) for the sake of consistency, thereby eliminating the different orders of mention in Mark.
Matthew describes the cloud that descended on the disciples as “bright” (Matthew 17:5). This is omitted in the parallel passage from Mark. Matthew’s addition of the word “bright” was probably inspired by “the glory of the Lord [that] appeared in the cloud” in the Exodus narrative (Exodus 16:10). Obviously, both Mark’s and Matthew’s Transfiguration text made it clear that God is present in the cloud that descends on the disciples, but Matthew made it even more explicit by describing the cloud as “bright.”