I'm a big user of Mathematica. Notebook concept origins from that indeed (which is a very useful concept, I must add). I experienced with maxima in the past and I think if I hadn't been using Mathematica at the time maybe I would have started using that.

What I like in Mathematica is that there are load of useful information available online and printed (yes, that's true for lot of other systems, too) and can be used almost all areas of Mathematica (at least at those areas where I'm interested in).

Above I read that Mathematica lacks linear algebra, abstract algebra and number theory. I'm not certainly sure about it since I've already solved both lin.alg and abstract alg. problems. I will definitely have a look in Sage's documentation what features it offers (there can be some, though... but to establish a standalone software because it would be missing from Mathematica, I don't know). Even the most sophisticated algorithms can be implemented easily in Mathematica.

Disadvantage is that you may need some time to get accustomed to that not to use loops and other C-like structures because Mathematica encourages one to use functional approach where you can solve almost everything with several powerful operators (or functionals as one could call).

Graphical opportunities are great and symbolic calculations are incredible (not necessarily about symbolic integrations and the like, but the symbolic approach how you can formulate a specific problem).

Again, a big disadvantage is the price. For me, the only one. However, companies often buy it (in my very neighbourhood there was a company who bought it with an additional package and that spared 4 engineer-months for them at the first time).

And what I don't like in Mathematica is the big hype around Wolfram and Wolfram Research.

But their soft still is great

If Sage addresses the same paradigm as Mathematica then it could evolve into a gem.