Do not forget that most of the exponential smoothing (ES) methods have underlying ARIMA model representations. For example simple ES corresponds to an ARIMA(0,1,1) model. An exception is multiplicative Winters method which can only be approximated by the additive Winters method on logs. What differs is that ES methods make use simple recurrences, at least for given smoothing constants (often roughly estimated), while ARIMA models require specification and estimation. A comparison is available in my paper with Laurence Broze.
Article Exponential smoothing: Estimation by maximum likelihood
Also note that the R forecast package combines powerful ES with ARIMA.