We've always thought about the mobile internet as a limited thing compared to the desktop internet, because of the constraints of hardware and network. Today, obviously, those constraints are a lot less than they were in the featurephone world, but it can still feel natural to talk of the PC as the most fully-featured version of the internet, and mobile as the place where you have to make lots of allowances for limitations of various kinds, just as for a smart watch. We make things 'mobile-first' because that's where the time and attention is, because that's where the use is and because mobile phones are in everyone's pockets and PCs (even laptops) are chained to desks, but it's still somehow a place with limits relative to the desktop.
I'd suggest that we should think about inverting this - it's actually the PC that has the limited, basic, cut-down version of the internet.
First, for 20 years the internet, on a PC, essentially meant a web browser, mouse and keyboard. There were other things happening around the margins, such as Skype and Spotify (and, for some people, email apps), but the web model was dominant. On a smartphone this is clearly no longer the case (which is why I sometimes call smartphones post-Netscape and post-PageRank). The interaction model is much more complex and sophisticated, and it continues to change all the time - Now, iBeacons, notifications, deep links, Apple Pay and Touch ID, keyboards and extensions and so on keep adding to and changing the options. Some of these new capabilities can get added back onto the PC, but many cannot, or arrive much later.
Second, a smartphone knows much more than a PC did. There's an old computer science saying that a computer should never ask you a question that it should be able to work out the answer to; a smartphone can work out much more. It can see who your friends are, where you spend your time, what photos you've taken, whether you're walking or running and what your credit card is. The sensors, APIs and data that are available (with permission - mostly) to a service you want to use on a smartphone are vastly greater than for a website isolated within a web browser on a PC. Each of those sensors and APIs creates a new business, or many new businesses, that could not exist on a PC.
The earliest and most obvious expression of this is the explosion of mobile social apps, almost none of which would have worked on a PC, yet which flourish on a smartphone because, as I frequently point out, the smartphone itself is a social platform - every app can see your friend list, access your camera, send you notifications and sit on the home screen two taps away from any other screen on your phone, so the friction of adopting them and of using more than one at once falls away.
But actually, one could generalize this beyond social - the smartphone itself is an internet platform in a way that a PC was not. On a PC the web browser was the internet platform, but on a smartphone it's the entire device and the browser is turned from 'the internet' to one icon, just a phone calls turned from the purpose of the device to just one icon.
That means that instead of thinking about the constraints of mobile - of the things you can't do because the screen is smaller and there's no keyboard - we should rather think of the PC as having the basic, cut-down, limited version of the internet, because it only has the web. It's the mobile that has the whole internet.