There is potentially semantic ambiguity in play here: in some fields onone uses dispersion in a narrow sense, meaning standard deviation. However, in general case standard deviation is only one of various possible measures of dispersion (some other measures of statistical dispersion are cited in Wikipedia article, already quoted in the other answer.)
This, in my perception, tells us the discrepancy between the smallest and largest value and does not measure the dispersion from the central value.
One obvious advantage of range in comparison to standard deviation is that it is not dependent on how one estimates the central value (using mean, median, mode or something else), which is important when we do not know the underlying distribution or when it is contaminated with outliers.
Remark
Speaking of semantics dispersion here really means statistical dispersion. Indeed, the term is routinely used in physics, biology, chemistry, and some other fields in completely different senses, totally unrelated to statistical properties. Another example of a seemingly obvious term that in fact causes much cross-field confusion is frequency.