Akali is the better of the both in terms of Assassin and duelist, whereas Diana is better as an initiator, and tank. Akali's kit flat out enables her to outplay almost anyone as long as she isnt hard cced. Diana does not have the escapes or vision tools that Akali has to outplay someone as an assassin often times. Her Lunar Rush isn't really a displacement but moreso a gap closer. Mark something and you can jump to it. You dont even have to mark something you just need vision, which is nice and works well with her Q, but it isnt enough to outplay many or use all that often.
Akali's R has more charges, her W hides her from normal vision (you most likely wont always be carrying a pink), and the biggest benefactor is her Spell vamp. Her kit is more bursty than Dianas and she can do it much faster, sustaining her health many times over and being a threat without the many Mana issues Diana has which is also a crippling factor for Diana. Diana isnt a pushover, she's just not all that efficient sadly, when it comes to the same role of assassination. If you player her on the more tanky/initiation side she's actually very good.
However I will say though that a Diana can beat an Akali in a 1v1, even if both have end game builds. Akali just has an edge due to the shroud. But if you hold on to your E as Diana and force her to pop it whilst having some W orbs up if and when you popped it, you can actually pull her into your shield and even kill her from that damage lol. I've done it quite a few times it's kinda funny. Invisibility=/=invulnerability.