Use Holt-Winters or ARIMA? My question is around the conceptual difference between Holt-Winters and ARIMA. 
As far as I understand, Holt-Winters is a special case of ARIMA. But when is one algorithm preferred over the other? Perhaps Holt-Winters is incremental and therefore serves as an inline (faster) algorithm?
Looking forward to some insight here.
 A: As Brian says in his answer: there's no simple rule as to which is better. For example, the UK's Office for National Statistics switched from HW to ARIMA and wrote a paper on it and while they chose to switch it was probably because of the power of the X12 (now X13) software package, which is ARIMA-based and very powerful, rather than the technique itself.
Also, you should compare State Space (Kalman Filter) solutions, which is even more general. R's arima, for example, uses a State Space solution under the hood.
Holt-Winters has three parameters, so it's simple, but they're basically smoothing factors so it doesn't tell you much if you know them. ARIMA has more parameters, and some of them have some intuitive meaning, but it still doesn't tell you much. State Space can be complex, but you can also explicitly model things for greater explanatory power. In my opinion, anyhow.
A: I have seen people with different data sets compare results from both algorithms and get different results.  In some cases, the Holt-Winters algorithm gives better results than the ARIMA and in others cases, it is the other way around.  I don't think you will find an explicit answer on when to use one over the other.
A: From what I have seen, ARIMA allows you to add independent regressors whereas Holt Winters does not provide that luxury
