FlashAttention is mathematically identical to standard attention, so in theory there's no downside. In practice, numerical inaccuracies of floating point mean that the results differ slightly. I don't know of any papers going in depth to analyze what impact those variances have in a range of real models, but generally speaking deep models handle slightly variances well. I've not noticed any difference in my applications training models. And tons of people use FlashAttention as a drop-in replacement on models trained on standard attention (e.g. using xformers in StableDiffusion).

Also in practice FlashAttention is still relatively new so it isn't well supported in libraries yet. Until PyTorch 2.0 you had to either implement it yourself, or use something like xformers which comes with a bag of caveats. PyTorch 2.0 now has it built-in, and it's easy to use, but the implementation is incomplete so you can't, for example, use it with an attention mask (which is needed in LLMs, for example).

tl;dr: Basically none, but it just isn't well supported yet.