As in, the thing you spend your time surfing – which term do you usually use? My impression is that “net” was more common before the dot-com boom, and that “web” has became more common since. I’m looking to y’all to confirm or deny. Pesonally, I find myself saying “web” more often these days.
Also, in your mind, do you distinguish between the internet and the world wide web? I can’t say as I really do myself.
I do, but then I am in IT. Inernet is the physical structure, world wide web is the media I access from the internet.
The internet encompasses more than just the WWW; usenet, email, gopher, and others. The terms are becoming blurred though; it usually requires context to determine what somebody is actually talking about.
The one I surf from Firefox is the Web; the internet includes the Web but also things like my VPN connexion to work or playing WoW.
I surf the Web but connect to the Internet, though I wasn’t aware of making such a distinction (nor did I know why I would) until this thread. 
On the rare occasion I actually refer to my internet activities as “srufing” I use the term intarweb. Mostly, I just say I am online.
The following things are “net” but not “web”:
Streaming audio or video feeds (RealMedia, QuickTime, etc)
Email
FTP, SFTP
Chat (IRC, AIM, MS Messenger, iChat, etc)
Usenet
File Sharing (Limewire, et. al.)
Skype and other telephony -over-TCP
Timbuktu, PCAnywhere, VNC, MS Remote Desktop, Apple Remote Access, etc
Filemaker Pro (fmp7:// networking)
SSH
RSS standalone clients
Apple Software Update (MS uses IE for Windows updates, yes?)
Various commercial programs auto-update routines
…and probably a dozen handfuls of other things I’m just not thinking of.
That’s what I always say, I’m online. Or whatever it is that I’m looking at is online, so “I was reading online that …” or “I saw the cutest video of a kitten online.”
It works as an adjective or an adverb, but now I’m wondering if it really makes sense … am I online, or is only the information online?
Unless you use web-based e-mail. 
Heh. Good point. Honestly, though, most of the morons with whom I have daily contact don’t have a clue what a computer is, let alone the capability to make a distinction like that, so it becomes moot.
Even then, the systems that send and receive the email are part of the net and not the web.
IMHO, the “average user” is somebody whose entire means of exposure to The Internet (“net” for short) is by way of an ISP and browser whose primary (if not only) means of coping with The Internet is by way of the hypertext-driven World Wide Web (“web” for short). He or she has little awareness of, and little interest in, the other aspects of The Net that are external to The Web. Thus, for the “average user” the terms are synonymous.
Those of us who know about the other realms on The Net, are, by my definitions, not “average users.”
YMMV
Using that interpretation of an “average user” it’s likely that at some point AOL and the internet were synonymous as well. Thankfully I think we’re past that now.
You may laugh (actually, I am laughing) but I’m having a REAL PHILOSOPHICAL CRISIS over this! When you read my response, it was information online - so I am online to you … but am I online to me? Is a flesh-and-blood person getting information from a computer screen separate from or a part of the internet?
Well, you could send yourself an e-mail. then you would be online to you. I bet that’s a relief.
(bolding mine)
Isn’t that “teh Inernet?”
Heh…I almost never laugh – just very few things are so out there as to make me more than smile. I just burst out laughing. Husband rushed in to make sure I was ok, it was such an unusual sound (not that I am uber-serious, I have fun, I just don’t often laugh).
I am so sorry to be the cause of a massive philosophical crisis…wait, no – that’s not that unusual for me…
You know, I work in a call center. One of the bits of information we try to gather is email address. I like to play “guess the aol’ers” I can 9 times out of 10 peg who has aol just from their general stupidity. But of course, the funniest thing I have come across yet, and I don’t know if the stupidity was fully the fault of the customer or the call center rep who entered the email address, but I came across an email that was whatever at americaonline.com. I fixed it without saying anything to the customer, but I was snickering in my sleeve for a few minutes.