What is the difference between the Shapiro–Wilk test of normality and the Kolmogorov–Smirnov test of normality? 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?
 A: You can't really even compare the two since the Kolmogorov-Smirnov is for a completely specified distribution (so if you're testing normality, you must specify the mean and variance; they can't be estimated from the data*), while the Shapiro-Wilk is for normality, with unspecified mean and variance.
* you also can't standardize by using estimated parameters and test for standard normal; that's actually the same thing.
One way to compare would be to supplement the Shapiro-Wilk with a test for specified mean and variance in a normal (combining the tests in some manner), or by having the KS tables adjusted for the parameter estimation (but then it's no longer distribution-free).
There is such a test (equivalent to the Kolmogorov-Smirnov with estimated parameters) - the Lilliefors test; the normality-test version could be validly compared to the Shapiro-Wilk (and will generally have lower power). More competitive is the Anderson-Darling test (which must also be adjusted for parameter estimation for a comparison to be valid).

As for what they test - the KS test (and the Lilliefors) looks at the largest difference between the empirical CDF and the specified distribution, while the Shapiro Wilk effectively compares two estimates of variance; the closely related Shapiro-Francia can be regarded as a monotonic function of the squared correlation in a Q-Q plot; if I recall correctly, the Shapiro-Wilk also takes into account covariances between the  order statistics.
Edited to add: While the Shapiro-Wilk nearly always beats the Lilliefors test on alternatives of interest, an example where it doesn't is the $t_{30}$ in medium-large samples ($n>60$-ish). There the Lilliefors has higher power.
[It should be kept in mind that there are many more tests for normality that are available than these.]
A: 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.
