I am an avid Mathematica user. Not because I need to evaluate that much math, but because I really like its wysiwyg notebook structure for note-taking. It's great for keeping up with lectures and including decent looking equations and tables when they come up (which is fairly often in technical classes). The export to latex is a nice added bonus.
As far as I know, there aren't great alternatives to the Mathematica front end - I'd love to be proven wrong, through.
part of what makes Mathematica notebooks particularly powerful is that they are Mathematica expressions themselves (Mathematica expressions are M-expressions), thus they can be thoroughly manipulated in all sorts of ways.
then there's Dynamic, which is the most straightforward event mechanism you're going to find in any language
Yes! I took all of my math notes in university in Mathematica notebooks. Lead to much quicker and deeper understanding of the material, not to mention the side effect that I then knew how to actually use the math to solve real problems.
One thing that came out of that was a nice Wheatstone Bridge calculator w/ GUI that was really useful for prototyping signal acquisition circuits.