In most registers of English, the reality is that the capitalized form Internet will not work. It's merely a common noun. The etymology and technical background being interesting in their own right, these are no more essential when to the point since in common parlance, such isn't and simply will never be known. The singular reason it will be capitalized in some circles is posts just like this; people just like you. Educated people trying to figure out what is right, what should be right, etc. In the higher registers of English, it'll remain a question where some people refer to what they've always seen, bewildered as to why they should need to capitalize it. That will mostly come from applications that correct people's mistakes.
The only reason I could see it as needing to be capitalized is that it could be perceived as a kind of institution or segment of human civilization that is very important, and the capitalization marks such. Look at it like the sun, the moon, and the Earth. I would say as well that as a foundational pillar of the Information Age, in academic writing, its capitalization marks its status for human society. However, there should be no real grammatical reason that it be capitalized and, like I said, I don't think it will be in the long run, most of the time.
You could find lots of distinctions between other nouns that serve different purposes, such as data in the singular or plural etc. That hardly merits capitalizing the one but not the other. It's not a company, isn't some department... I think we would like for it to be capitalized (those who are reading) just because it's a stylistic addition and we intuit that it enjoys a very, very special status in the grand scheme of human inventions. Still, even from that perspective, rockets, phones, cars; they're all mere common nouns. In short, English is an economic language, not an academic one. Short and simple tends to win the race. Applications like Word will eventually be updated, eliminating the question for 99% of people who would have otherwise thought to ask.