Riddle is not fully formed yet. He's a memory almost made flesh, so he can grab Harry's wand and even create some shiny letters with it, but it's possible that's the extent of his powers.
Riddle also knows that the Killing Curse didn't work on Harry once and rebounded on Voldemort, so he might be wary of trying this powerful spell again on Harry, especially in his state.
Riddle wants to humiliate Harry first, to show him that his powers are no match to his. As he says:
So this is what Dumbledore sends his great defender. A songbird and an old hat. Let's match the power of Lord Voldemort, Heir of Salazar Slytherin, against the famous Harry Potter.
And in the book, this is preceded by:
I’m going to teach you a little lesson.
Just like in the graveyard duel in Goblet of Fire, Riddle\Voldemort favors a spectacle, a feat of strength, to show Harry that he's not the great Gryffindor hero everyone says he is. Because...
Out of Universe Reason
...Voldemort is the very epitome of a supervillain, and as such, is prone to gloating, monologuing, explaining his evil plan to the hero and then putting him in some death trap that always fails to work. When you simply kill the hero, you are just a killer... But when you mock him first and plan his demise in some fiendishly clever way, you're truly a villain. ;)