The eigen-vectors of $U$ are $b_0,\ldots,b_{n-1}$ with eigen-values $e^{2\pi ik/n}$, $k=0,\ldots,n-1$. Hence $(U-1)b_k=e^{2\pi ik/n}b_k-b_k=(e^{2\pi ik/n}-1)b_k$, i.e. the eigen-values of the difference operator are $e^{2\pi ik/n}-1$, $k=0,\ldots,n-1$.