HoltWinters() implements exponential smoothing along the lines you will find in any description, e.g., Wikipedia or Forecasting: Principles and Practice. Note how smoothing parameters and initial values are chosen (
The function tries to find the optimal values of alpha and/or beta
and/or gamma by minimizing the squared one-step prediction error
if they are ‘NULL’ (the default).
For seasonal models, start values for ‘a’, ‘b’ and ‘s’ are
inferred by performing a simple decomposition in trend and
seasonal component using moving averages (see function
‘decompose’) on the ‘start.periods’ first periods (a simple linear
regression on the trend component is used for starting level and
trend). For level/trend-models (no seasonal component), start
values for ‘a’ and ‘b’ are ‘x’ and ‘x - x’, respectively.
In contrast, quoting from
ses, holt and hw are simply convenient wrapper functions for
ets() fits a state space model, as per FPP2 and Hyndman, R.J., Akram, Md., and Archibald, B. (2008) "The
admissible parameter space for exponential smoothing models".
Annals of Statistical Mathematics, 60(2), 407-426. One difference therefore is how the transition matrices in the state space model (which correspond to the smoothing constants) are chosen and how the initial values are set.
An additional difference is that state space models will allow calculating predictive densities and prediction-intervals, in contrast to the earlier ad-hoc exponential smoothing methods (which are forecasting methods, not statistical time series models, although point forecasts may be the same).