I think the difference is that distribution free statistics use limit theorems to show that you can construct tests of the weak null hypothesis about some function of the distribution function that are asymptotically consistent and unbiased. I think there is some deficiency in the popular understanding of what constitutes "nonparametric" statistics, and that the statement of the null hypothesis eludes many applied statisticians.
For instance, if a 2-sample test has data drawn from a family of parametric probability models $X_1, X_2, \ldots, X_n \sim_{iid} \mathcal{F}_{\theta_1}$, $Y_1, Y_2, \ldots, Y_m \sim_{iid} \mathcal{F}_{\theta_2}$, then that parameter forms the obvious test of the weak null hypothesis: $\mathcal{H}_0: \theta_1 = \theta_2$. However, there might be a functional value which, independent of the distribution, requires an asymptotically correct test: so with two samples having no known class of probability models $X_1, X_2, \ldots, X_n \sim_{iid} \mathcal{F}$, $Y_1, Y_2, \ldots, Y_m \sim_{iid} \mathcal{G}$, you define a "parameter" for that distribution: $\theta_1 = f(\mathcal{F})$, $\theta_2 = f(\mathcal{G})$, and test the weak null hypothesis: $\mathcal{H}_0: \theta_1 = \theta_2$. In most cases, limit theorems such as the $\delta$-method, central limit theorem, etc. can show that there exists test statistics for which $n^k g \left( X, Y \right) \rightarrow_d \chi^2_p(\lambda)$ where the non-centrality parameter $\lambda$ is 0 only if the weak null hypothesis is true. There are usually some regularity conditions on the distributions $\mathcal{F}$, $\mathcal{G}$, but in general they can apply to a large array of probability models. The statistic $g()$ is what we call distribution free. Take the T-test: for normal, exponential, negative binomial, and poisson data, the T-test is well powered to detect differences in the means between these distributions for any two-sample experiment, even when the mean does not parametrize that distribution.
non-parametric statistics are traditionally used when researchers wish to state their strong null hypothesis as $\mathcal{H_0}: \mathcal{F} = \mathcal{G}$. That is, if there is a difference in the 99.99th percentile in these data (and even if there is a symmetric difference in the 00.01th percentile), they would like a test that is calibrated to reject the null hypothesis. We don't wish to prespecify what we would believe to be a meaningful difference in the distribution for these data, whether it be a mean, median, or quantile based summary. This is stupid in my mind. Tests like this can be done with Kolmogorov-Smirnoff tests for the empirical distribution function or using non-parametric smoothed kernel density estimates. The required sample size to detect the egregious, symmetric quantile difference above is enormous, but it is powered to reject the strong null hypothesis in that case.
Rank based statistics like the log-rank, Wilcoxon, Mann Whitney U, etc. are not tests of the strong null hypothesis, so in my mind they're not non-parametric.
