I read from a few sources that certain exponential smoothing methods (linear state-space) have an equivalence form as ARIMA. For example, simple exponential smoothing (SES) as ARIMA(0,1,1) and Holt's linear exponential smoothing as ARIMA(0,2,2). I was wondering, however, if each pair of the models are equivalent, why did I get different results from software packages (e.g. SPSS) when using, say, SES and ARIMA(0,1,1) to fit the exact same set of data. If SES is truly a "special case" of ARIMA(0,1,1), shouldn't SES always perform slightly worse (or at most on par with, at optimality) than ARIMA? I observed that SES gave a better fit on several data sets, compared to ARIMA(0,1,1).
Does it have anything to do with the stationary assumption?
What is the true definition of the state-space form of exponential smoothing? Does it simply mean the exponential smoothing model with stationarity and normally distributed, zero mean, independent error terms assumptions? Does it require anything else?