The utility of Renyi entropies?
Renyi entropy is analogous, in some sense, to $\ell_p$-norms, so let's first recall why those norms are useful.

Suppose we have a vector of numbers $a \in \mathbb{R}^n$. We want to have a single number that represents, in some sense, how does the typical element of $a$ look like.

One way to do so is to take the average of the numbers in $a$, which roughly corresponds to the $\ell_1$ norm: $\mathbb{E}_{1 \le i \le n}[|a_i|]$. This is often useful, but for some applications it has the following problems: First, the $\ell_1$ norm does not give us a good upper bound on the largest element of $a$, because if there is a single large element and many zeroes, the $\ell_1$ norm will be significantly smaller than the largest element. On the other hand, the $\ell_1$ norm also does not give us a good bound on how small the elements of $a$ are, for example, how many zeroes $a$ has - this problem occurs in exactly the same scenario as before. 

Of course, when the elements of $a$ have a lot of variance, such as in the extreme scenario as above, no single number can give solve both problems above. We have a tradeoff. For example, if we only want to know the largest element, we can use the $\ell_\infty$ norm, but then we will lose all information about the smaller elements. If we want the number of zeroes, we can look at the $\ell_0$ norm, which is just the size of the support of $a$.

Now, the reason for considering $\ell_p$ norms is that they give us the whole continuous tradeoff between the two extremes. If we want more information on the large elements we take $p$ to be larger, and vice versa.

Same goes for Renyi entropies: Shanon's entropy is like $\ell_1$ norm - it tells us something about the "typical" probability of an element, but nothing about the variance or the extremes. The min-entropy gives us information on the element with the largest probability, but loses all information about the rest. The support size gives the other extreme. The Renyi entropies give us a continuous tradeoff between the two extremes.

For example, many times the Renyi-2 entropy is useful because it is on the one hand close to Shanon's entropy, and thus contains information on all the elements on the distribution, and on the other hand gives more information on the elements with the largest probability. In particular, it is known that bounds on the Renyi-2 entropy gives bounds on the min-entropy, see, e.g., Appendix A here: http://people.seas.harvard.edu/~salil/research/conductors-prelim.ps