What is the difference between the Shapiro-Wilk test of normality and the Kolmogorov-Smirnov test of normality? When will results from these two methods differ?
Briefly stated, the Shapiro-Wilk test is a specific test for normality, whereas the method used by Kolmogorov-Smirnov test is more general, but less powerful (meaning it correctly rejects the null hypothesis of normality less often). Both statistics take normality as the null and establishes a test statistic based on the sample, but how they do so is different from one another in ways that make them more or less sensitive to features of normal distributions.
How exactly W (the test statistic for Shapiro-Wilk) is calculated is a bit involved, but conceptually, it involves arraying the sample values by size and measuring fit against expected means, variances and covariances. These multiple comparisons against normality, as I understand it, give the test more power than the the Kolmogorov-Smirnov test, which is one way in which they may differ.
By contrast, the Kolmogorov-Smirnov test for normality is derived from a general approach for assessing goodness of fit by comparing the expected cumulative distribution against the empirical cumulative distribution, vis:
As such, it is sensitive at the center of the distribution, and not the tails. However, the K-S is test is convergent, in the sense that as n tends to infinity, the test converges to the true answer in probability (I believe that Glivenko-Cantelli Theorem applies here, but someone may correct me). These are two more ways in which these two tests might differ in their evaluation of normality.