Even if Twilio were simply providing a proxy for accessing the Google Alerts API, I would use Twilio's proxy over Google's.
Twilio consistently provides excellent documentation and developer support, whereas Google's results on that front are very mixed (varies widely by product).
That's a bit of a stretch. You wouldn't get better results than using Google's APIs, if anything you introduced another failure point and authentication credentials that you need to track and update regularly.
You don't need to use Google at all. This would be a 20-minute project using the HN API.
Twilio is a whole extra level of cruft. Any machine can send an email, so unless you want a voice call to tell you someone mentioned you on HN it makes absolutely no sense.
Twilio consistently provides excellent documentation and developer support, whereas Google's results on that front are very mixed (varies widely by product).