Yes and no -- of course if one uses more points the result can be more accurate. The real difference between a center difference and contour integration is tha

Yes and no -- of course if one uses more points the result can be more accurate. The real difference between a center difference and contour integration is that differences suffer from loss of information in finite-precision arithmetic. For example, in (f(x+h) - f(x-h))/(2h), the numerator actually contains very few bits of information when h is small. The Cauchy integral formula, in contrast, behaves quite well so long as the point of evaluation is bounded away from the contour.