Actually, in the solar system gaps between planets range from 1.4 to just under 2, if you count the failed planet where the asteroid belt is (probably got shredded by Jupiter's gravitational influence while it was trying to form). This tends to happen because 1.4 or so is roughly the region of stability for coplanar elliptical near-circular orbits; any closer and the third body perturbation significantly messes around with the orbit over a long time scale.
Lunar systems of gas giants are also governed by orbital resonances and stability. Typically however for moons you get PERIOD resonances, where the periods of two orbits are nearly integer ratios of each other (ex: Ganymede, Europa, and Io are in a 1:2:4 resonance). Period scales with the 3/2 power of orbital radius, so you can pretty quickly calculate that while gas giant moons fall into the same rough regime (1.4-2x the distance of the closest neighbor), in practice you get moons even closer than you'll get planets because the 3rd body perturbations of the sun are even more apparent.
Capsule description: Same overall mechanism governs spacing between planets and moons, but the 3rd-body influences are sufficiently different to affect the observed behavior.