In which cases should one prefer the one over the other?
I found someone who claims an advantage for Kendall, for pedagogical reasons, are there other reasons?
I found that Spearman correlation is mostly used in place of usual linear correlation when working with integer valued scores on a measurement scale, when it has a moderate number of possible scores or when we don't want to make rely on assumptions about the bivariate relationships. As compared to Pearson coefficient, the interpretation of Kendall's tau seems to me less direct than that of Spearman's rho, in the sense that it quantifies the difference between the % of concordant and discordant pairs among all possible pairwise events. In my understanding, Kendall's tau more closely resembles Goodman-Kruskal Gamma.
I just browsed an article from Larry Winner in the J. Statistics Educ. (2006) which discusses the use of both measures, NASCAR Winston Cup Race Results for 1975-2003.
I also found @onestop answer about Pearson's or Spearman's correlation with non-normal data interesting in this respect.
Of note, Kendall's tau (the a version) has connection to Somers' D (and Harrell's C) used for predictive modelling (see e.g., Interpretation of Somers’ D under four simple models by RB Newson and reference 6 therein, and articles by Newson published in the Stata Journal 2006). An overview of rank-sum tests is provided in Efficient Calculation of Jackknife Confidence Intervals for Rank Statistics, that was published in the JSS (2006).
I refer the honorable gentleman to my previous answer: "...confidence intervals for Spearman’s rS are less reliable and less interpretable than confidence intervals for Kendall’s τ-parameters", according to Kendall & Gibbons (1990).
Again somewhat philosophical answer; the basic difference is that Spearman's Rho is an attempt to extend R^2 (="variance explained") idea over nonlinear interactions, while Kendall's Tau is rather intended to be a test statistic for nonlinear correlation test. So, Tau should be used for testing nonlinear correlations, Rho as R extension (or for people familiar with R^2 -- explaining Tau to unsuspecting audience in limited time is painful).
Here's a quote from Andrew Gilpin (1993) advocating Kendall's τ over Spearman's ρ for theoretical reasons:
"[Kendall's $τ$] approaches a normal distribution more rapidly than $ρ$, as $N$, the sample size, increases; and $τ$ is also more tractable mathematically, particularly when ties are present."
Gilpin, A. R. (1993). Table for conversion of Kendall's Tau to Spearman's Rho within the context measures of magnitude of effect for meta-analysis. Educational and Psychological Measurement, 53(1), 87-92.
FWIW, a quote from Myers & Well (research design and statistical analyses, second edition, 2003, p. 510). If you still care about the p-values;
Seigel and Castellan (1988, nonparametric statistics for the behavioral sciences) point out that, although $\tau$ and Spearman $\rho$ will generally have different values when calculated for the same data set, when significance tests for $\tau$ and Spearman $\rho$ are based on their sampling distributions, they will yield the same p-values.