We analyze the Swift/BAT sample of short gamma-ray bursts, using an objective Bayesian Block procedure to extract temporal descriptors of the bursts' initial pulse complexes (IPCs). The sample is comprised of 12 and 41 bursts with and without extended emission (EE) components, respectively. IPCs of non-EE bursts are dominated by single pulse structures, while EE bursts tend to have two or more pulse structures. The medians of characteristic timescales–durations, pulse structure widths, and peak intervals–for EE bursts are factors of ∼2-3 longer than for non-EE bursts. A trend previously reported by Hakkila and colleagues unifying long and short bursts–the anti-correlation of pulse intensity and width–continues in the two short burst groups, with non-EE bursts extending to more intense, narrower pulses. In addition, we find that preceding and succeeding pulse intensities are anti-correlated with pulse interval. We also examine the short burst X-ray afterglows as observed by the Swift/X-Ray Telescope (XRT). The median flux of the initial XRT detections for EE bursts (∼6x10–10 erg/cm2/s) is ≳ 20x brighter than for non-EE bursts, and the median X-ray afterglow duration for EE bursts (∼60,000 s) is ∼30x longer than for non-EE bursts. The tendency for EE bursts toward longer prompt-emission timescales and higher initial X-ray afterglow fluxes implies larger energy injections powering the afterglows. The longer-lasting X-ray afterglows of EE bursts may suggest that a significant fraction explode into denser environments than non-EE bursts, or that the sometimes-dominant EE component efficiently powers the afterglow. Combined, these results favor different progenitors for EE and non-EE short bursts.