I'm teaching myself probability theory, and I'm not sure I understand any use for variance, as opposed to standard deviation. In the practice situations I'm looking at, the variance is larger than the range, so it doesn't seem intuitively useful.
In practice, you calculate the SD through calculating the variance (as abutcher indicated). I believe the variance is used more often (apart from interpretation, as you indicated yourself) because it has a lot of statistically interesting properties: it has unbiased estimators in a lot of cases, leads to known distributions for hypothesis testing etc.
As to the variance being bigger: if the variance were 1/4, the SD would be 1/2. As soon as your variance/SD are smaller than 1, this order reverses.
In portfolio theory, variance is additive. In other words, just as the return of a portfolio is the weighted average of the returns of its members, so to is the portfolio variance the weighted average of the securities' variances. However, this property does not hold true for standard deviation.
Variance is the most basic of the two measures... stddev = sqrt(variance). While exaggerated, it's good enough for a comparison and grows very large when there is mixed-up-ness in the distribution.
variance(22, 25, 29, 30, 37) = 32.3 variance(22, 25, 29, 30, 900) = 152611.0
Standard deviation is used way more often because the result has the same units as the data, making standard deviation more appropriate for any sort of visual analysis.
I think you have to really qualify your question when you refer to the practical use of the variance. For instance, in business there's no practical use for the variance. The standard deviation has more of a practical use by giving a mathematical representation of variation that can be understood and applied. For instance, the standard deviation can be used to quantify risk as indicated in the calculation of the Beta for a stock. The variance has no practical application comparable with the standard deviation. If we move to higher level statistical analysis then the variance has many practical applications, but only when dealing with higher level analysis, which is not the focus of the vast majority. So it truly depends on the area in which one may be a practitioner. For business practitioners, the only use for the variance is to find the standard devotion.