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.