Hi @Cat ,
This is a great question and I'm sure people will have some different opinions, but here's a quick version of mine.
First and most importantly is that these are different measures of alpha diversity, so therefore there is no reason to assume one should be significant if the other is also significant. What you choose should be based on your biological question.
The Chao1 is a species richness estimator which unlike Species Richness, tries to estimate unobserved taxa by assuming a Poisson distribution of the data. The estimates are quite sensitive to how your data has been processed (including rarefying depth) and how you've filtered rare taxa (think singletons/doubletons). I never quite understood why it is used with typical microbiome data because with typical workflows that include rarefying and removing rare species, you essentially first remove those rare taxa, and then with Chao1 you try to re-estimate them back in?
There's also other assumptions like that all the species have an equal chance of being observed, which also doesn't usually hold in most microbiome surveys.
Shannon diversity on the other hand is an index, that takes into account the abundance of the species, weighting in rare species, but doesnt try to estimate unobserved diversity. It is a lot more stable than Chao1 when it comes to rarefying depth and you can usually see a plateau in rarefaction curves with very few sequences. This makes it - in my opinion- more resistant to processing bias and thus a bit more reliable.
tldr; they answer different questions, Shannon is rather robust while Chao1 (in my opinion) doesn't make a whole lot of sense with microbiome pipelines.
If you want to estimate unobserved richness, instead of Chao1, I would recommend using Amy Willis' breakaway package for which there is a q2-plugin. Also her group has done some pretty awesome stuff on the topic and I recommend reading some of her work for example here, and here.
Hope this helps.