Which bound is better, a logarithmic or a polynomial with arbitrarily small degree?
Since $\log n = O(n^{\epsilon})$ for an $\epsilon &gt; 0$, if you can prove an approximation ratio of $O(\log n)$, then approximation ratios of $O(n^{\epsilon})$ (for any $\epsilon &gt; 0$) immediately follow. You should always prove the best approximation ratio that you can, unless:


The best approximation ratio holds only in expectation, and some other approximation ratio holds with high probability.
You have several incomparable approximation ratios (this is more common in expressions involving more than one parameter).


That is, your worse guarantee needs to have some advantage over your better guarantee for you to report both.

Regarding practical performance, you are highlighting the fact that asymptotic performance can be misleading when it comes to evaluating algorithms in practice. The most well-known instance is probably "fast" matrix multiplication, which is usually slower than the trivial algorithm. Here you have two options:


Prove non-asymptotic guarantees on the approximation ratio, say $100\log n$ and $(2/\epsilon)n^{\epsilon}$. This allows you to obtain concrete guarantees for every $n$.
Do experiments. The experiments reveal the actual approximation ratio on the average. If you can fit your results to a nice function, you can say that empirically, the average approximation ratio is (say) $10\log n$, though in the worst case all you know is (say) $100\log n$. Experiments, however, are not so welcome in theoretical papers, unfortunately.