You would disable it through the language and input settings. It's easy to disable. Speech Input and then Google Settings. You'll see the OK GOOGLE recognition setting if you want to turn it off. Or retrain it if it stops working, too.
It's very handy while traveling. You can do it through OK GOOGLE or the Maps app, and just say the name or address. It will basically take you there. That's the best thing.
Beyond that, it's just easier than opening mobile Chrome. And you can launch most apps with it. Some apps are optimized for voice (Google) and you can say "make appointment" and then just say the time and place, or "start sleep" if you can't be bothered to just hit the Sleep As Android widget (I use the widget), or ask it absolutely anything that someone asks you. "How old is so-and-so?" And an immediate answer. "Closest pizza restaurant" and so on. You get used to it. "Call Dr. So-So." And it does it.
I don't think I will use Cortana like that because it's a PC. And Siri often says "I didn't understand that" on my iPad. (It was a freebie; I buy Android for mobile. But I do have access to IOS apps.)
Haven't tried Cortana, but it will need to go a long way to beat Google.
You do sacrifice privacy, but I figure we lost that a while ago. In for a penny, in for a pound.
rjm
Sent via my most phabulous phablet, the Galaxy Note 4