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.