Solved – When is distance covariance less appropriate than linear covariance

I have tried to collect a few remarks on distance covariance based on my impressions from reading the references listed below. However, I do not consider myself an expert on this topic. Comments, corrections, suggestions, etc. are welcome.

The remarks are (strongly) biased towards potential drawbacks, as requested in the original question.

As I see it, the potential drawbacks are as follows:

1. The methodology is new. My guess is that this is the single biggest factor regarding lack of popularity at this time. The papers outlining distance covariance start in the mid 2000s and progress up to present day. The paper cited above is the one that received the most attention (hype?) and it is less than three years old. In contrast, the theory and results on correlation and correlation-like measures have over a century of work already behind them.
2. The basic concepts are more challenging. Pearson's product-moment correlation, at an operational level, can be explained to college freshman without a calculus background pretty readily. A simple "algorithmic" viewpoint can be laid out and the geometric intuition is easy to describe. In contrast, in the case of distance covariance, even the notion of sums of products of pairwise Euclidean distances is quite a bit more difficult and the notion of covariance with respect to a stochastic process goes far beyond what could reasonably be explained to such an audience.
3. It is computationally more demanding. The basic algorithm for computing the test statistic is $O(n^2)$ in the sample size as opposed to $O(n)$ for standard correlation metrics. For small sample sizes this is not a big deal, but for larger ones it becomes more important.
4. The test statistic is not distribution free, even asymptotically. One might hope that for a test statistic that is consistent against all alternatives, that the distribution—at least asymptotically—might be independent of the underlying distributions of $X$ and $Y$ under the null hypothesis. This is not the case for distance covariance as the distribution under the null depends on the underlying distribution of $X$ and $Y$ even as the sample size tends to infinity. It is true that the distributions are uniformly bounded by a $chi^2_1$ distribution, which allows for the calculation of a conservative critical value.
5. The distance correlation is a one-to-one transform of $|rho|$ in the bivariate normal case. This is not really a drawback, and might even be viewed as a strength. But, if one accepts a bivariate normal approximation to the data, which can be quite common in practice, then little, if anything, is gained from using distance correlation in place of standard procedures.
6. Unknown power properties. Being consistent against all alternatives essentially guarantees that distance covariance must have very low power against some alternatives. In many cases, one is willing to give up generality in order to gain additional power against particular alternatives of interest. The original papers show some examples in which they claim high power relative to standard correlation metrics, but I believe that, going back to (1.) above, its behavior against alternatives is not yet well understood.

To reiterate, this answer probably comes across quite negative. But, that is not the intent. There are some very beautiful and interesting ideas related to distance covariance and the relative novelty of it also opens up research avenues for understanding it more fully.

References:

1. G. J. Szekely and M. L. Rizzo (2009), Brownian distance covariance, Ann. Appl. Statist., vol. 3, no. 4, 1236–1265.
2. G. J. Szekely, M. L. Rizzo and N. K. Bakirov (2007), Measuring and testing independence by correlation of distances, Ann. Statist., vol. 35, 2769–2794.
3. R. Lyons (2012), Distance covariance in metric spaces, Ann. Probab. (to appear).