Yeast plays an important role but from my experience it is not as pronounced as it is in brewing. In brewing you can easily taste the difference between an american ale yeast and a british ale yeast- and you can easily tell the difference between a higher fermentation temp and a lower one.
With distilling I have noticed that the distilling process removes most of the yeast flavor. I still don’t use turbo yeast as I have had really bad results with it. I use bread yeast for my rum as that works great. I use US05 brewers yeast for my single malts, and I have been mixing it up with my corn mash.
If you pitch a good amount of healthy yeast, don’t ferment too hot, ferment all the way out- you will get good results. Find a yeast or two that work well for you, or just keep experimenting.