It all boils down on how would you want to process a time series as to breakdown its components as to use these for later prediction or classification.
For one ARIMA is a parametric method (assumption of a fixed distribution) modeling a stationary time series based on static ARMA terms while with wavelets you model a wavelet function by selecting a list of characterictis you want the function to have as to best approximate a signal (wavelets can model a non stationary as well as stationary). In wavelets the length of the filter the number of vanishing moments and the symetry of the mother wavelet vs the signal will define how good the function is in providing representation of time resolution and frequency resolution. (existing a trade off between this two terms as you get shorter filters you get better in time resolution while decrease the frequency resolution.
In ARIMA you aproximate the signal components by selecting ARMA terms from acf and pacf while for wavelets you aproximate the signal by selecting a mother wavelet, filter length and number of vanishing moments. (the number of decomposition levels provides capturing certain frequencies per level). So an stationary ARMA model would have its pseudo equivalent for a non stationary signal by performing a wavelet transform only at one scale obtaining scaling and detail coefficients (sort of equivalent to having an ARMA with dynamic terms). The results for the two methods will be different, for as in Arima you have static ARMA terms across time, in wavelet decomposition you will have frequency ranges across time which will not correspond to the ARMA terms, although you could calculate what are the frequencies that the ARMA terms represent and approximate these by using wavelets, the dynamic approach of the wavelet will ensure that there is no one specific frequency across time where the ARMA terms will be represented).
In modeling a time series the first issue is to define the characteristics of the signal and the second is in knowing in what part of the signal we are most interested in. (modeling the entire signal or extracting a portion of it).
Which modeling method will provide a better representation of the signal or portions of it you are interested in, depends on the ability of the modeler, the properties of the signal (how easy is to model it) Sometimes an stockastic model such us ARMA or other modeling method model will be sufficient, other times using a built in transform by selecting carefully its terms will give better representation.