Interpretation of advanced composition theorem of differential privacy
Consider the composition of $k$ algorithms each of which is $(\varepsilon,0)$-differentially private. We want to calculate parameters $\varepsilon'$ and $\delta'$ such that that the composition of these algorithms satisfies $(\varepsilon',\delta')$-differential privacy.

The comparison is between advanced composition
$$\varepsilon' = \sqrt{2k\log(1/\delta)}\varepsilon + k \varepsilon (e^\varepsilon-1)$$ and basic composition $$\varepsilon' = k \varepsilon$$.

There is no way to say that one of these expressions is universally better than the other, as it depends on the parameter settings. However, we can argue that for some realistic parameter settings the first is better than the second.

What do I mean by realistic parameter settings? Let's say that the number of queries is fairly large (e.g., $k = 10^5$) and the final privacy parameters are not too big (e.g., $\varepsilon' = 1$ and $\delta'=10^{-6}$). 

Then for basic composition we need $\varepsilon = \varepsilon'/k = 10^{-5}$.
For advanced composition the calculation is a bit messier, but we can calculate that $\varepsilon=58 \times 10^{-5}$ suffices. In other words, advanced composition saves a factor of $58$ over basic composition. The improvement becomes more dramatic as $k$ increases.

Asymptotically, if we think of $\varepsilon'$ and $\delta'$ fixed but $k$ becoming larger, we see that basic composition requires $\varepsilon \approx 1/k$, while advanced composition requires $\varepsilon \approx 1/\sqrt{k}$. (I'm dropping all non-relevant terms and constants here.) This is the advantage of advanced composition.