What is the importance of global phase? How it affect a unit vector if we see it on a Bloch sphere? What is the meaning of 'up to global phase' in exercise 4.3 of Nielsen and Chuang?
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The point of the phrase "up to a global phase" is that a global phase is not important. It has no observable consequences.
So, states $$ |\psi_1\rangle=a|0\rangle+b|1\rangle,\qquad |\psi_2\rangle=e^{i\phi}(a|0\rangle+b|1\rangle) $$ are the same up to a global phase. There is no experiment that you can do to distinguish them. This is represented in the Bloch sphere by the fact that both states map to the same point.
The the N&C exercise, it's the same issue, but for unitaries rather than states: $U$ and $e^{i\theta}U$ have the same effect on a state, except for a difference in global phase, which is irrelevant. So it doesn't matter which of the two I make. Make whichever is easier. (Caveat: controlled-$U$ and controlled-$e^{i\theta}U$ are very different.)
answered Sep 16, 2020 at 6:48