Most of these questions are pretty straight-forward
Both fit into two roles well.
Irelia top/jg
Diana mid/jg - diana JG has become much harder to use effectively though with the new jungle hitting as hard as it does.
(edit, I suppose Irelia could fit a support role as well in low elo, however definitely wouldn't recommend it)
Both have a dash, Diana's Q/R gives her a free dash from R to a moonlighted target, Irelia's Q gives her a resetting dash if it kills her target.
Irelia has true sustain along with true damage on her W.
Diana's W gives her a shield with damage but no true sustain
Both have a form of CC - diana's E is an AoE pull coupled with a slow, Irelia's is a slow or stun based on comparative % health. (ex, Irelia at 50% health, enemy at 52%, she stuns. Reverse %s and she slows - this essentially ensures she doesn't have a stunlock when also winning trades)
You have to bear in mind they hold very different roles.
Diana serves as an assassin who's able to stick to her targets particularly when caught out, a great pick into
for example.
Irelia serves best as a bruiser with her
core, can dish out quite a lot of burst damage with just that as well as being able to take a fair amount of damage.
For me, I like Diana. She's just funner to use and isn't nearly as unforgiving as Irelia.