How can genetically identical twins look different?

There are two key pieces of terminology that it is important to understand with this question; genotype and phenotype. All sexual organisms inherit a mix of genes from their parents. This mix - the genetic constitution of an organism - is what we call the genotype, and in identical twins this mix is the same in both individuals. How an organism actually looks is described by what we call the phenotype -  this is a product of the genotype, and an organism's interaction with it's environment. Identical twins do not grow up in  identical environmernts - they may only be subtly different, or drastically - but never identical, unlike their genes. Extreme examples of this may be physical deformity as a result of accidents - scars, for example. Subtle differences accumulate throughout life - every conversation, film watched and book read influences an individual's life. Imagine identical male twins -  one twin a passionate rower, another a talented musician. You could expect differences in the phenotype of these twins' physiques - the rower may have broader, more developed shoulders, and less dexterous hands compared to his musical twin.

These twins carry the same genes, but they are expressed differently based on their lifestlyes. Expression is how we describe the what parts of the whole genotype appear in the phenotype. Epigenetics is the mechanism explaining how environment can affect the phenotype - it explains how your brain cells and skin cells carry the same DNA, but look completely different. In short, factors from the environment can cause small molecules can join onto the top of DNA, or onto the proteins it winds around. This changes the way DNA is read, meaning different genes are more or less likely to be expressed.