I am building a mobile app (android et al) that I need to use SMS for. The SMS would simply send an activation code when user first registers with my app (and maybe one other equally simple usage). Since Google is a respected company and Google voice is free (API), I was thinking of going with Google Voice. However, a lot of developers seem to be using Twilio. Is there an advantage to using Twilio SMS over Google Voice SMS for my basic usage of sending verification code (again, Google Voice is free)?
The API you are linking to is not an official Google API and therefore there is no guarantee that either it will work reliably or that Google will not decide to disable the API at some point. So you are paying for a level of reliability rather than relying on an unofficial hook into Google Voice.
There are lots of SMS APIs available, Twilio is a well known one for sure but there are others. Take a look at another answer I posted a while back listing some of the providers. How to send SMS programatically in a professional and reliable way?