Yes, they have the same DNA. Super pedantic people will argue that you can in fact pick up a tiny number of acquired differences between a cell from twin A and a cell from twin B. However, that number is the same as you would get comparing two different random cells from the same twin.
For all intents and purposes, by any rational measure, the twins' DNA is identical. Epigenetics is a completely different thing, and evidence in how that works functionally in humans is almost totally lacking still.
Personality traits are influenced by much more than simple genetics. The human brain is the most complex thing known to exist, and is shaped hugely by environment, as well as by random factors.