What makes one better or worse may depend on many things.
Ultimately (though this is not especially enlightening) - if you want to look at the kind of thing IQR measures, IQR is good at that, while if you want to look at the kind of thing standard deviation measures, standard deviation is good at that. The different ways they respond to data helps determine where they're more useful (in particular, sd is much more impacted by large outliers, so tends to be more valuable in situations where the data doesn't have big tails)
I want to give a slightly different take that may or may not be of some help.
Both IQR and standard deviation can be thought of as measures of a kind of "typical distance between data points".
For example, the IQR is effectively the distance between the median of the top half of the data and the median of the bottom half of the data, and in that sense is a kind of 'typical distance'.
The variance (up to a Bessel correction factor in the sample case) is half the average squared distance between pairs of points; the standard deviation is thereby a root-mean-square distance between pairs of points, divided by $\sqrt{2}$ (or times $\sqrt{\frac{n}{2(n-1)}}$ in samples when applying a Bessel correction in the sample case.
The different way those 'typical distance' measures are affected by observations in different parts of the data is useful; it may be worth exploring an empirical influence function (without necessarily naming it, the concept is intuitive enough).
These lines of discussion may not be suitable for every level of student, but it's a take on two common measures of spread that may be easier to motivate that some others, and has at least one advantage: it reduces the size of that 1.35 "asymptotic conversion factor" at the normal to the perhaps less surprising 0.954 (or 1.048, depending on which way you go) - that's a good deal closer to 1, and it's smaller for the measure less affected by very large observations.