If you never write the words ado or adieu, this isn’t even going to matter. If that’s you, I’ll forgive you for back-paging with haste. Now, without further ado(?), let me get to the point.
While both versions are pronounced the same, and both are correct, they mean different things.
The word ado references a fuss made of something unimportant; dragging a thing out for no good reason. What you’re saying is, “let’s get on with it,” whatever it may be. Compare that to adieu, which is French for goodbye!
Well, now I know.
Usage
To indicate that we should move forward and continue:
…and without further ado, let me introduce the star of the show: you!
…or to indicate that it’s time to depart:
…and without further adieu, I leave you, my love.
So, both ways are correct! But, only if you use them correctly.
Yes, I’m a geek.