In short, Hangouts will not use your call minutes. It does use data, so may eat up your data plan if you're not on Wifi. It's a fairly efficient protocol, so you may not notice the hit to your data cap.
Wow, even "in short" is too long....
Google Voice and Hangouts are two separate things stuck together, with a little bit of GooglePlus paste. It makes things a bit confusing, since they both do "voice and text communication".
Google Voice was a purchased product, Grand Central. Grand Central grew up before smartphones and the cell plans back then were very different than we know today. As such, it's very phone number centric, and there are features to optimize costs based on rules that are no longer current, so the features make less sense today.
Think Grand Central/Google Voice as "product for geeks to do cool things for dumbphones" It did for phones what every programmer wants to do always; it created an abstraction layer, or in English, a virtual manipulable phone number. For incoming calls, that meant: trying any of possible multiple "real" phones to find you, call screening, and voicemail. For outgoing calls, it allowed calls from any of your "real" phones to look like it came from your virtual (this was clunky, oy!) and also helped with long distance - you called a "local" number and typically got better rates than from your phone company. Also, when you made a call, you were calling "cell" numbers out, back when carriers price differentiated on that. Or you could have outgoing calls look originate by a call to YOUR phone, from your virtual number. This helped when you could get the "5 friends I talk to for free" or whatever plans. All of this kind of clunky and confusing. I juggled all this, because it was worth it for me to have a personal virtual number mapped to my work phone, and then had only one device to carry.
For Google Voice, to make an outbound call on a smartphone, you'd go into the Google Voice app, tell it to call some number. Google Voice would contact Google servers, and (depending on settings) Google would call you and make the outbound call, or you would dial out (to yet another virtual number, hosted at Google). THIS IS WHERE THE VOICE MINUTES YOU READ ABOUT CAME FROM. All these calls used cell minutes, either inbound or outbound.
Now, Hangouts was invented as a Skype competitor. To really compete, it needed to be your messaging app. SMS? Sure, lets bake that in, and tightly integrate with both SMS and Google Plus, two very different beasts. What could possibly go wrong? Skype out? Well, we can do that. Lets have data only, VoIP dialing to POTS lines, and lets use the Google Voice number.
So, now my phone has a Google Voice app and a Google Hangouts app. Both can make calls out, in different ways, looking like the same number. Both can make SMS out. Google Voice has SMS as data out, but incoming can come in as "true" SMS as well. How can that be confusing? Oh, and Incoming SMS for "classic" Google Voice came from a "virtual Google phone number" not from your friend, so you could text back to the virtual number and they'd get from your Google Voice number. But.. incoming calls always came from their real number, unless you configured it to come from *your* virtual number... As I write this out I marvel at how I kept this in my head straight for so long. My Phone Book is littered with "GVoice" entries, the virtual numbers that every contact you get an SMS from or dial out to gets mapped by Google. Confused yet?
So, Google Voice still does the normal phone line juggle, so uses cell minutes (this is what you asked about). Google Hangouts uses VoIP, probably using WebRTC. It's a new protocol, and last time i checked, most VoIP apps (including Asterisk) haven't been able to connect with it. On iOS/iPhone, outgoing calls on Hangouts are VoIP only, no audio minutes. Incoming calls are through normal cell service. And all of this can change without notice. Confused? :)