Your DNA is like a datacenter with a billion nodes that all start out with the same content in DRAM. Although error correction can compensate for most bit flips, at this scale you still get a really low mean time between failures, leading to differing memory contents over time.

According to Wikipedia, "In human cells, both normal metabolic activities and environmental factors such as radiation can cause DNA damage, resulting in as many as 1 million individual molecular lesions per cell per day."

There are mechanisms to repair this damage, but they don't always work. Sometimes the result is cancer.

An excerpt from my genetics professor:

"Even in the absence of mutagens, an average gene will mutate about once in a million generations. ~10^16 cell divisions occur in a typical life span so somewhere in cells that are part of you each of your genes has mutated 10^10 times (hence cancer)."