The answers here that stress that "the answers here that stress convenience are missing the crucial point" are missing a crucial point. There are at least two ways to approach variance (which are sort of dual to one another):

(1) I need to measure the spread of a distribution of values. What measure should I use? <i>[Possible answer: variance.]</i>

-or-

(2) If I define V[X] = E[(X-E[X])^2], then V[X] has many nice properties and seems to relate well to other parts of the theory and even other parts of mathematics. Obviously, it's something pretty important theoretically. What are its practical uses? <i>[Possible answer: as a measure of spread.]</i>

If the only thing you care about is measuring spread, then convenience may be the *only* reason for you to use the variance, if you use it at all. I don't think anybody can seriously claim that among all measures of spread, the variance is absolutely the best-quality measure in all situations. Yes, the variance is additive and allows you to formulate the central limit theorem, and properties like that are certainly very nice to have, both in theory and in practice, but they don't make it a better measure of spread. So from this perspective, "convenience" often is the right answer. 

On the other hand, as a theorist, you would probably want to develop the theory along the most fruitful route, so you would be stupid to ignore the variance in any case. Its usefulness as a measure of spread then is less important, and its overall properties are rather more important. From this perspective, "convenience" isn't really the right answer, since it doesn't convey the intrinsic value it has by virtue of the excellent theory surrounding it.