Since the particles in a liquid move freely but are still relatively close together, how does that change the thickness of the liquid?

The particles in a liquid are much closer together than in a gas, and are free-flowing. The thickness of a liquid, which is called viscosity, is caused by attractions between particles. A liquid that's very viscous doesn't necessarily have particles that are closer together or moving slower, but it does have stronger attractions between the particles. Water is more viscous than pure ethanol because water molecules are more attracted to each other. This also explains why ethanol evaporates faster than water.

You've probably noticed that when ice melts, the liquid that forms doesn't have an observable difference in viscosity from water at a higher temperature. Pure substances melt at a precise temperature and go abruptly from solid to liquid rather than slowly softening and thinning.

Mixtures of substances tend to change viscosity with a temperature change. For example honey, which is an aqueous solution of a sugar, becomes less viscous when heated. Most oils do the same, because they're a mixture of hydrocarbon molecules. This happens because the presence of a different substance interferes with the ability of one substance to crystallize.