Here’s a video from WinBeta that shows what Cortana will be like in Windows 10.
She can do all the things her windows phone version can but in this instance, the desktop Cortana does not have access to the same databases so she cannot respond with the usual witty replies or connect to the calendar.
This is just an early version.
BTW, the more I use Android, one of the things I learn to appreciate more about WindowsPhone is just how good Cortana actually is as a personal voice assistant. I hate trying to use Google Now for any voice queries (a few of the useful stuff Cortana does, or cannot do). So this could definitely be a strong point of differentiation, especially if Microsoft put extra effort into pushing development in the AI.