Is it, from an educational perspective, still sound advice to recommend people to use paper/a notebook (of the traditional sort, not the device) and pen/pencil?
I wonder if computers are a disadvantage since whatever program you use, it will be giving you clues when it comes down to writing math symbols (so, you'll be thinking less). Add to it the fact that most examinations will be without a computer.
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I use just about every device known to man to do math: pencil and paper, chalk and white boards, tablet with stylus and computer. They all have their advantages and disadvantages, but I would have to agree with you that one major disadvantage of any internet-connected device is instant access to immediate help, all the time. Students - myself included - have gotten into really bad habits when it comes to learning mathematics because of this. So my vote would be yes: pen, pencil, chalk, dry erase or possibly a tablet (with the WiFi turned off...) is probably the best way to actually learn mathematics. It's completely different once you start doing research, of course - then a computer with LaTeX and a CAS and the internet becomes absolutely indispensable. I also use my tablet for reading and annotating papers (which is great).
I would also agree that at the high school or early college level, a well designed online tutoring/homework system can be good. But again - it's an internet connected device, and I've had way too many students who do their online homework with Wolfram Alpha open in another tab...so who knows.
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Pen and paper allows quick revisions of ideas. The computer helps ensure you don't make stupid mistakes on the little details when trying to see the big picture. In my opinion, there are times and places for both tools.
Often when I find myself lost while working on the computer, prying myself away and sketching the problem out on a piece of paper is the only way to break out of it. Often I discover within minutes that I had been overlooking something obvious, because the computer tends to make me impatient and unwilling to calculate basic details.
On the other hand, sometimes when I'm in the lab with a pen and a pad and I find myself stuck and thinking in circles, all it takes is plugging some equation into Mathematica and calculate a few points for me. ("Oh, right, it diverges. That's why I couldn't prove it converges.")
The real way to learn about math is to realize there are parts of it which comes some ways and parts which come others. There's no one way to do it, and probably no one way will work for any one person. It's about understanding; if you don't understand, try something else.
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With regard to note-taking: do what is most practical for you. Much of the educational benefit of note-taking is tactile memory - your memory is enhanced by the physical action of note-taking. (The remainder of the benefit is, of course, having a reference to look back on...provided you take reasonably readable notes. If you have neat handwriting, develop a strict organizational system for your notes as well - it will pay off handsomely.) So pen-paper, tablet-stylus, don't really make much of a difference when you get down to what's important. Every style has its distinct advantages and disadvantages for note-taking, which other have elaborated on.
That said, whatever you choose, learn to write with it efficiently. Too often, students spend too much time with their heads down, trying to write down as much as they can coming out of the lecturer's mouth. Learn to balance between writing and really, truly listening - listening not just with your ears, but with your eyes and all your other senses. A handy skill is being able to write without looking at the paper for short periods, and for this I recommend blank paper and clipboard rather than lined paper or a notebook.
Typing your notes as you hear them in lecture is something I don't really recommend, though I won't speak out against it. I find typing notes for math much slower than direct pencil or stylus manipulation (especially these days when I'm dealing with more commutative diagrams), so I would end up directing too much time and attention to my notes and too little to the lecture. Typing up notes is for after the lecture, if I have the time (though I prefer to write them by hand again, just neater). Also, some lecturers dislike laptops being open during lecture, and the sound of typing may inconvenience the lecturer and your fellow students.
With regard to homework, problem-solving, etc: a flexible system is best. I prefer blackboard, then paper-pencil; tablet-stylus doesn't appeal to me for scratch work, because I can't crumple a tablet and toss it in the bin. Blackboards provide a wide workspace, accomodate many pictures, and erase quickly; paper-pencil lack the last advantage for me. For collaborative work, nothing beats the mighty blackboard.
With regard to teaching: I am a firm believer in the old blackboard lecture. Lectures using powerpoints and other devices invariably make me fall asleep. Blackboard lectures have a few distinct advantages in that department over powerpoint lectures. One, having to write everything forces the lecturer to slow down and work carefully, vital to catching mistakes in the lecture notes and keeping the pacing reasonable. Two, blackboards allow for improvisation, something that can't be done easily with powerpoint. Three, a speaker using the blackboard is visually much more animated and stimulating than a speaker standing in front of a podium reading a powerpoint. Projectors and transparencies are the same story. Other newfangled devices that they're apparently cramming into every high school classroom nowadays are no better. I don't mean to say the computer should be completely divorced from lecture, but relying on them too much makes teaching quality deteriorate.
My university switched over to using an online program called ALEKS. It is an artificial intelligence program that helps determine the exact point of confusion in solving a problem. EXAMPLE: I student might get a system of equations question wrong, not because they don't understand how to solve these systems, but because they don't fully understand how to manipulate fractions.
ALEKS has the user solve a couple different equations of similarity and uses that to determine what step the student is struggling on.
Systems such as this are becoming more and more popular and will continue to become more advanced.
I think the answer to your question is that Computers are the most efficient way to teach Math, but paper and pencil are still the most effective for TESTING.
:)