Yup, different. The White Walkers (called “Others” in the books) are essentially necromancers initiated while human by the Night’s King, and can be dispatched by dragonglass or Valyrian steel. Not sure if actual dragons’ fire can kill them, as dragons are magical or at least harbingers of magical activity in the series. Regular fire I don’t believe is given the chance to harm Walkers because their chilly aura snuffs out non-magical combustion.
The Wights are the undead raised by the Walkers. They can be killed by fire or, if memory serves, total bodily dismemberment.
The term “walkers” seems to be used metonymically, as the host of undead almost certainly outnumbers the actual White Walkers.