You seem to be mixing context and content. Pretty much all your content should be available for mobile and desktop although there are exceptions in certain situations, such as if you provide a heavily location based service on mobile.
However there is definitely a mobile context which drives the way you prioritise and deliver that content for mobile. Assuming that just because the user is on a mobile device, then the user is actually mobile is a non starter - 20% of mobile use is on the couch. Making device based decisions about content inclusion is tantamount to mind-reading. You cannot and should not do it.
Quoting Luke Wroblewski:
Just because my screen is small doesn’t give anyone insight to my
behaviours, desires and needs.
You can tune the presentation and hierarchy of information on mobile devices, but the presence is valuable in both cases. Absolutely you can use responsive design to accomplish this.
Much of this answer so far comes from a brief interview Luke Wroblewski gave to Joe Welinski of BlinkUX, of which you can read my rough version of the transcript.
Having said that, it's important to really understand your mobile users and design for them rather than designing differently for mobile just for the sake of it.
As something of a clue to the difference between mobile and desktop context and yet the need for similarity in content, there is an increasing importance in being able to provide a cross channel experience where you can resume a task on desktop, that you started on mobile (or the other way around, but that's less common).
Read more on that by Karen McGrane on A List Apart where she ends with:
It’s time to stop imagining that smartphones, tablets, and desktops
are containers that each hold their own content, optimized for a
particular browsing or reading experience. Users don’t think of it
that way. Instead, users imagine that each device is its own window
onto the web.
The desktop context is this: You have huge amounts of space on a static screen (or more than one screen) where you have the full attention of the viewer, in a comfortable environment, and with a 99.99 percent uptime on a mega-fast broadband connection – and a keyboard and mouse to boot.
The mobile context is this: You have more constraints: Smaller screens, patchy or dynamic signal coverage, fat fingers, lower performance, limited data plans, attention span (data snacking), location and posture. But you also have more capabilities: orientation detection, dynamic location detection, compass, accelerometer. How many wifi hotspot providers don't think about the mobile context when giving you a free wifi signup form?
In addition to the constraints and capabilities that the mobile context provides, you also have different behaviours. There are three types of user behaviour on mobile - what Google termed urgent now, repetitive now, and bored now, and depending on the purpose of the website you should align your design with the target behaviour. Don't confuse these behaviours with the more focused user needs and desires for a given website/app.
Now here's the thing: the line between desktop and mobile is getting fuzzier all the time - some desktops are starting to encroach on the mobile space with removable touchscreen displays and the like. And as I mentioned above, there is an increasing case for cross channel tasking. The couch use case is when I'm too lazy or it's inconvenient to go to my desktop, but a website that is too awkward to use on a mobile device, or that is missing content is going to force me to move to desktop. We've all been there right?
It's not that there isn't a mobile context and a desktop context - it's just no longer a binary state.
I go into the desktop-to-mobile ux in more depth, with examples in my slides/notes on the mobile user experience.