We design efficient distance approximation algorithms for several classes of structured high-dimensional distributions. Specifically, we show algorithms for the following problems: - Given sample access to two Bayesian networks $P_1$ and $P_2$ over known directed acyclic graphs $G_1$ and $G_2$ having $n$ nodes and bounded in-degree, approximate $d_{tv}(P_1,P_2)$ to within additive error $ε$ using $poly(n,ε)$ samples and time - Given sample access to two ferromagnetic Ising models $P_1$ and $P_2$ on $n$ variables with bounded width, approximate $d_{tv}(P_1, P_2)$ to within additive error $ε$ using $poly(n,ε)$ samples and time - Given sample access to two $n$-dimensional Gaussians $P_1$ and $P_2$, approximate $d_{tv}(P_1, P_2)$ to within additive error $ε$ using $poly(n,ε)$ samples and time - Given access to observations from two causal models $P$ and $Q$ on $n$ variables that are defined over known causal graphs, approximate $d_{tv}(P_a, Q_a)$ to within additive error $ε$ using $poly(n,ε)$ samples, where $P_a$ and $Q_a$ are the interventional distributions obtained by the intervention $do(A=a)$ on $P$ and $Q$ respectively for a particular variable $A$. Our results are the first efficient distance approximation algorithms for these well-studied problems. They are derived using a simple and general connection to distribution learning algorithms. The distance approximation algorithms imply new efficient algorithms for {\em tolerant} testing of closeness of the above-mentioned structured high-dimensional distributions.