Google Speech is the API Google uses in their iOS apps. iOS devs can use the API in their own apps. Google does have multiple implementations of speech to text (STT) for on device (off line) processing but usually that is used in combination with their objectively better online (Google server) processing. Its get a bit sketchy though when it comes to pricing. If you use Google Speech API they charge you less if you allow google to log your users speech data.....and since app development is a business design to make money and paying less overhead = more profit....well...
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Review pricing for Speech-to-Text
cloud.google.com
Much like Apple they claim its to improve the speech to text. This is Googles wording so my bias doesn't rub off,
keep in mind though thats for the developers to opt in and out with their app not the user...
cloud.google.com
Apple has their Speech Framework available to iOS devs. Its "free" if you ignore the yearly expense for being a dev and Apple tax. To be fair Apple uses everyones voice to "improve their product". However Apple requires devs to be very explicit that the STT is going to be used for product improvement.
Google on the other hand skirts around this requirement by giving the user instructions on how to turn the mic on in settings with no word that their voice is being logged or anything...here is a comparison.
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Also if you use Apples Speech you can only ask the user that question one time, subsequent request are blocked. Apple thinks if you bug someone enough they will eventually say Yes...probably right..more info on that..
Ask the user’s permission to perform speech recognition using Apple’s servers.
developer.apple.com
I should mention that Google is asking for generic microphone permissions while Apple is asking specifically for permission for the app to use their Speech Recognition framework. Google just wants the mic on and its just assumed its to use their Speech API, which is a fair assessment IMO.
Apple and Google aren't the only game in town though. There are a half dozen companies that develop STT SDK's for iOS for high quality on device (offline) use.
I prefer Apple, it seems to work better for me which might just be because I've adjusted too it from using it for so long. I can say they all suck in their own way, so does Alexa. Privacy concerns aside I think what it mostly comes down to is how well the STT handles your language and your accent. Me having a standard east coast American accent both of them do fairly well with Apple picking up my voice over other people and background much better. Oh and it can identify me, vs other people which is nice for HomeKit.
Man...that was long winded...apologies...