I've been using RWS's Language Weaver and DeepL's stock engine in parallel for about a year now. Thought it would be interesting to share my experiences.
For complex sentences without defined terminology, DeepL is still performing better than LW. (I did not make use of DeepL's custom dictionaries as I use Studio 2021 and don't have the DeepL subscription level that would be required.)
For simple to average sentences with defined terminology and an established tone of voice, LW outperforms DeepL. The time savings are significant.
(The third MT I use is OpusCAT MT, which is free, adaptable and runs on my machine. It often outperforms DeepL with simple sentences or phrases where it had sufficient training data. It's free - no cost at all, so I hesitate to compare.)
The ability to train LW and use the dictionary for defined terminology is a winner. Marketing and related texts tend to avoid overly complex sentences or phrases. For this use case LW is ideal and a huge time-saver.
This is not the result of a controlled study, just my observations from a year of using both.
Daniel