if they don't/won't do,then why does some youtube downloader apps thrown out of the PlayStore (Tubemate, Vidmate etc ..)?
if they do, why there are plethora of web sites/apps doing still doing it?
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It's all about the ratio of impact level / easiness to take them down. In other words, worthiness of bothering.
A mobile app will have a bigger impact and is much easier to throw out of the Playstore compared to websites which will have smaller audiences and are more difficult to take off the web. If Youtube learns that any particular website offering downloads is causing enough revenue loss, surely they will care of it.