Variance/standard deviation versus interquartile range (IQR) Variance and interquartile range (IQR) are both measures of variability.
But IQR is robust to outliers, whereas variance can be hugely affected by a single observation.
Since variance (or standard deviation) is a more complicated measure to understand, what should I tell my students is the advantage that variance has over IQR?
 A: The main use of variance is in inferential statistics.
So, variance and standard deviation are integral to understanding z-scores, t-scores and F-tests.
This means that when your data are normally distributed, the standard deviation is going to have specific properties and interpretations. When your data are not normal (skewed, multi-modal, fat-tailed,...), the standard deviation cannot be used for classicial inference like confidence intervals, prediction intervals, t-tests, etc., and cannot be interpreted as a distance from the mean.
You can say things like "any observation that's 1.96 standard deviations away from the mean is in the 97.5th percentile." if your data are normally distributed.
A: There are several advantages to using the standard deviation over the interquartile range:
1.) Efficiency: the interquartile range uses only two data points, while the standard deviation considers the entire distribution. If you are estimating population characteristics from a sample, one is going to be a more confident measure than the other*.
2.) Meaning: if you data is normally distributed, the mean and standard deviation tell you all of the characteristics of the distribution. The interquartile range doesn't really tell you anything about the distribution other than the interquartile range.
3.) If you are willing to sacrifice some accuracy for robustness, there are better measures like the mean absolute deviation and median absolute deviation, which are both decent robust estimators of variation for fat-tailed distributions.
4.) Better yet, if you distribution isn't normal you should find out what kind of distribution it is closest to and model that using the recommended robust estimators.
*It's important here to point out the difference between accuracy and robustness. Standard deviation is never "inaccurate" per ce, if you have outliers than the sample standard deviation really is very high. "Outliers" usually means either data that you're not certain is legitimate in some sense or data that was generated from a non-normal population.
TL;DR don't tell you're students that they are comparable measures, tell them that they measure different things and sometimes we care about one and sometimes we care about the other. Tell them to think about what they are using the information for and that will tell them what measures they should care about.
A: A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range1. The standard deviation and mean are often used for symmetric distributions, and for normally distributed variables about 70% of observations will be within one standard deviation of the mean and about 95% will be within two standard deviations(68–95–99.7 rule). For non-normally distributed variables it follows the three-sigma rule.
As shown below we can find that the boxplot is weak in describing symmetric observations. 

Generated by this snippet of R code(borrowed from this answer): 
set.seed(1)
normal <- rnorm(10000)
a_vector <- c(-3, -2.65, rep((-2:2)*.674, 5), 2.65, 3)
boxplot(normal, a_vector)

We can see that the IQR is the same for the two populations 1 and 2 but we can see the difference of the two by their means and standard deviations. 
mean(normal); var(normal); mean(a_vector); var(a_vector)

-0.00653703946166382
1.02486558733286
-3.0626842058625e-17
1.95567142857143

We can see from the above case that what median and IQR cannot reflect can be obviously conveyed by the mean and variance. 
References:
1. https://en.wikipedia.org/wiki/Standard_deviation
