A "price breakout" usually refers to a market moving away from a previously established price range. A "volatility breakout" can mean either: (a) a market is trading with a higher daily/weekly/monthly price range than normal or (b) the implied volatility on the options is moving higher, probably in concert with a price breakout.
A volatility breakout is a sudden dramatic price movement: i.e. 0.8*ATR, or 3*ATR, etc.. A price breakout could be either a volatility breakout, or a breakout through a price level: i.e. support, resistance, channel edge.