Similarities:
Both involve ion movement. Both use ion channels to move ions across the cell membrane, in or out of the cell.
Differences:
Passive Transport (or Diffusion) moves ions from high concentration to low, using no metabolic energy. Active Transport moves ions from low concentration to high, using metabolic energy in the form of ATP. Active Transport uses an ion pump (or Sodium/Potassium pump) to move 3 sodium ions out of the cell and 2 potassium ions in to the cell through the energy of 1 ATP molecule.
In active transport particles move against a concentration gradient and therefore require energy which must be supplied by the cell. Carrier proteins that are found in the cell membrane of cells use energy to transport molecules or ions across the membrane, against the concentration gradient.
Passive transport is a naturally-occurring phenomenon and does not require the cell to exert any of its energy to accomplish the movement. In passive transport, substances move from an area of higher concentration to an area of lower concentration.