Forecasting vs non-forecasting predition for time series anomaly detection
I had the same question and ultimately tried an LSTM forecaster and LSTM autoencoder. For me, the choice of forecaster vs autoencoder had effects on the following two points:

Seasonal Pattern:
if you have a seasonal pattern (e.g. weekdays vs weekend) in your data, and the timeframe you are looking at has the pattern -- but is shifted by e.g. a day, you are likely to find that outlier with the Forecaster, but not as likely with Autoencoder.
This Disadvantage of the Autoencoder can be circumvented by also feeding in the weekday and time for each datapoint.

Needed Datapoints:
With the Autoencoder, you don't need datapoints before the timeframe you want to examine. You can just take the series you are interested in and give it to your model.