Two early side-by-side comparisons of Cortana with Siri and Google's Voice Search/Now contend (and demonstrate) that the Microsoft assistant achieved comparable performance:
It thus appears that Microsoft has taken away Siri or Google's assistant capabilities as competitive differentiators vs. Windows Phone. Indeed in the Gizmodo test Siri lagged in a few cases.
Neither review says that Cortana has exceeded the other assistants at this point. But the fact that Microsoft is out of the gate with a comparable capability is impressive. The only major thing that now stands in Windows Phone's way is its more limited app selection.
Let's hope that Cortana now puts pressure on Apple to further upgrade Siri. Since its dramatic introduction nearly four years ago Siri has not lived up to its potential, though it has continued to improve incrementally.
Google Voice Search performs adequately. It's speech recognition for dictation is consistently not as good as Apple's (Nuance's). Google Now's "anticipatory search" and related features are much more interesting. However Google Now also hasn't evolved significantly in the past 12 to 18 months.