Personal experience on my part. I'm Android all the way but my wife & a number of her friends us iPhones of varying vintage. On multiple occasions this issue has arisen where iPhone to iPhone text were fine but any sent FROM iPhone TO Android simply vanished into cyber space. In every instance, the culprit was found to be that the senders had iMessage enabled but SMS was disabled - enabling SMS (and) iMessage solved the problem. Personally, I cannot fathom why Apple clings to the antiquated iMessage system - far as I can determine, serves no useful purpose and only creates headaches.
The purpose of iMessage is for people who don't have unlimited messaging plans - their iPhone to iPhone messages are sent without using sms, so they don't count against the plan. You can surely turn iMessage off and sms on and it works fine - though my wife's was set the other way (Apple seems to reset iMessage back on when you update the os without telling you, being the pains in the *** that they can be.) And, if course, like Google's hangouts, iMessage messages are a lot more secure than sms. IMessage messages are encrypted end to end with each message encrypted so only the recipient can read it.
If anything it's sms that's antiquated. As well as Apple's reluctance to port it to Android or Windows phone which would be the best solution and is what Google sort of did with the very similar hangouts service (though no hangouts for you, Windows phone.)