This happens even from just spontaneous mutations. The body has many copies of DNA and they are not in sync with each other. Your cells diverge from your twins' cells, but also from each other.

Not just twin, but your former self too.

The DNA of every single cell in your body can have spontaneous mutations. But what is important for what you become in general is the DNA of your first cell that is rapidly multiplied to form your body. And what is important for what you leave on the next generation, is the DNA of your reproducing cells, which can too be individually altered, of course.