Two different questions here.
1) Can a small form factor mobile computer work as an effective replacement for a full fledged PC?
2) Is the Note 2 that device?
The answer to the first one for me is a definite, although qualified yes. I've had an N800 for about 5 years now and I took it on a three week backpacking trip and it functioned well at the time as my only computer. I was able to check and respond easily to emails, use Skype for calls, read and edit Office documents, have access to full internet sites, including sites specific to work. During this time, if I had a good wifi signal, I was able to work and do probably 80-90% of what I needed without feeling like I needed to have a real computer.
The Note 2 is undoubtedly a more powerful machine, and all in all it is the more capable device...but using it I don't get the same feeling. It feels much harder to write anything longer than a paragraph on the device, and switching between apps isn't as seamless, and browsing the internet feels more compartmentalized, where if you come across a video on some sites it might open up a different app to play it rather than have it just inline. All in all it feels like using a device that is running an OS that was designed to be an appliance as opposed to a full featured desktop replacement...and for me, until that changes the Note won't really feel like a PC replacement.