Logarithmic vs double logarithmic time complexity
Do not forget that $\log n$ still grows exponentially (in $\log(n)$) faster than $\log(\log n)$!

Indeed, if you look at the quotient of $\log(n)$ and $\log(\log(n))$, there is not much impressive to see:


[source]

But still, you get a factor five to six for sizes up to $100000$. Note that larger sizes are not uncommon in practice, and a speedup by that factor is awesome! It may make the difference between having results after lunch or only tomorrow. Be aware that part of the speedup may be eaten away by higher constants of the tree implementation; you would have to plot (or analyse) $c\cdot \log(n)$ and $d\cdot \log(\log(n))$ with $c,d$ the actual runtime constants to get a real picture.

Additionally, what Dave mentions is important: if the operation sped up thusly is executed, say, linearly often, constant speedups become linear speedups, i.e. you may decrease the leading constant of your entire algorithm! As I said above, that is awesome. Just look at what happens if you run the operation $n$ times:


[source]

Now if that's not worth the trouble I don't know what.