All diploid cells have the same set of DNA. What makes each cell different

Ultimately what makes each cell different despite the fact that they share the same set of DNA is regulation of gene expression, which regulates the expression of genes. In other words, it determines which genes are turned off and which are turned on in a cell. Gene expression can be regulated in many ways; it's mainly centered around molecules like transcription factors that have the ability to turn certain genes on and off and activators, which promote transcription of DNA.

The short answer to this question is cell differentiation. Differential gene expression results from the genes being regulated differently in each cell type, and differential gene expression leads to different cells.

In fact, from the very beginning of our lives, so to speak, when we're developing from a zygote into an embryo, the egg that makes up the zygote in fact already has a sequential program of gene regulation tat is carried out as cells divide, and this program makes the cell become different from each other in a coordinated fashion.