We consider whether the variability properties of Gamma-Ray Bursts (GRBs) that produce bright optical and longer wavelength transient afterglows (A-GRBs) are the same as a larger, inclusive sample of bright, long-duration GRBs, selected only by their gamma-ray emission. This sample may include a significant population of physically distinct ``dark or ``faint-afterglow GRBs with different variability properties, or may be composed of a single population, some of which lack afterglows only because of observational selection effects. We argue that the structure function is the most appropriate method for measuring the variability of bursts because of their transient and aperiodic nature. We define a simple statistic: the ratio of the integrated structure function from 0.1 to 1 s compared to that from 0.1 to 10 s, as measured in the observer frame. To avoid instrumental effects we restrict our analysis to GRBs with BATSE data. Comparing 10 A-GRBs to a ``main sample of about 500 bursts, we find there is a probability of only 0.03 of the samples being drawn from the same population, with the A-GRBs tending to have relatively less power on sub-second timescales. We conclude that this result is tentative evidence for variations in the properties of GRB progenitors that affect both the gamma-ray and afterglow properties of long-duration GRBs. In addition, our method of analyzing variability identifies a characteristic timescale of ~1 s, below which variability is suppressed, and finds a trend of increased short timescale variability at higher gamma-ray energies. The long-duration GRBs that we identify as having the most sub-second timescale variability, may be relatively bright examples of short-duration GRBs.
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