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The exponential smoothing method gives us values like SSE and $R^2$ for the entire model. The ARIMA model, however, does not give us these values.
So, given the same data, how do one decide which model is better? If we have to use numerical (not graphical) analysis to decide, that is.
Why doesn't ARIMA give SSE or $R^2$? I believe it does. You have the original data, the fitted values and model residuals, which are enough to calculate both of these quantities.
However, note that comparing $R^2$ will always favour more flexible models regardless of whether they are more or less relevant. You can always achieve an $R^2=1$ by fitting a polynomial of order $T-1$ to a sample of size $T$, but will that be a "good" model?
Better think of out-of-sample performance measures (functions of out-of-sample forecast error), information criteria or the like. (But beware that, say, AIC cannot be directly compared between an exponential smoothing model and an ARIMA model. You will need to adjust the AIC values to account for the fact that the ARIMA model effectively uses a smaller sample than the exponential smoothing model; Rob J. Hyndman has a post on that here.)
