DNA of identical twins
DNA is not the only thing in biology that determines a phenotype (observable trait, e.g. fingerprint). Far from it.

Traits can be affected by the environment (a very good example, temperature-dependent sex determination), stochastic (random) phenomena - these are know to occur in transcription ("reading DNA") or the decay of mRNA (messenger molecules which instruct protein building) - and by epigenetic phenomena. The degree to which DNA is actively read, or silenced, does not wholly depend on the DNA. In fact, this depends on the exact state of the cell or nucleus. Roughly speaking, cells in your brain (e.g. neurons) and the cells that line your lungs have identical DNA, but they are shaped, look, behave and react differently to the outside world. This is due to epigenetics, think of it as the life history of the cell.

Here's an analogy: reading the exactly same book can have different effects on people. The book is the DNA, its information unchanging, but the person and their exact status are the complicated world of the environment and the cell that is using its DNA for some purposes. PS. Terrible analogy but it may help.

By no means is my list exhaustive. Hope you can see how that identical twins with identical DNA can have a different shade of skin color (tanning!), different fingerprints (subtle and complex differences during digit development), or a different sense of fashion (psychology, cognition, what have you).