Soft water contains lower levels of calcium and/or magnesium than hard water.

Hard water is higher in dissolved minerals, (largely calcium and magnesium) than soft water

It is really soft water (due to salt based water softeners) that is the problem because it triggers acne due to the higher sodium levels.  This can be managed with a potassium-based softening system.  

Though hard water isn't great either, it doesn't trigger it the way a water softener can.