Authenticator works even when you don't have any sort of network available for your smartphone.
I don't know about your mobile provider, but I don't trust mine to deliver SMS messages in anything that resembles a timely manner.
Beyond that, it is more secure, as you've noted.
Even to activate the Authenticator, first you need to complete SMS/Voice setup.
Then, follow the directions for your type of device explained on Google Help page for 2-Step Verification.
Read this Make Your Email Hacker Proof for more understanding.
I recently got a similar message, "Your Google verification code is" and then an unusually long verification code (much more than the normal six characters you get through 2 factor authentication).
An hour or so before, I had used the Hangouts app and tried to verify my phone number. The first attempt failed, and for unrelated reasons I restarted the phone ...
Google Voice requires another phone number, so there is no point to that. I created a Skype number for this purpose. However, there is a fee required.
You could theoretically let that subscription lapse, but the next person to get the phone number would get your codes, although I think it would be nearly impossible for them to know which account the codes ...
Google now supports 2-Factor Authentication through a process called U2F. Yubico have made a series of Yubikeys that now support the U2F standard.
At this time the only browser capable of supporting U2F authentication appears to be Google Chrome.
SMS is the most common way to deliver OTPs. It’s convenient as almost everyone has a phone so they can easily get SMS. At the same time, SMS is considered as one of the least secure methods to get an OTP due to the risk that SMS messages may be intercepted or redirected. NIST is no longer recommending two-factor authentication systems that use SMS, because ...