How are points awarded?

Some of my opponents get for their wins about 20 points. Their rating is about the same size as mine. But I've never got such a large number of points for my wins. Why?

You can check the rating system of lichess here: lichess.org/faq#ratings
But I think the answer to your question has to do with the rating deviation, basically if a player hasn't played for quite a bit of time then they might have improved and their rating might not represent their real strength. So compared to a player that plays regularly (meaning that the rating is more likely to represent their actual strength) they might win (and lose) more points for a win (or loss).

Please if I am wrong on that feel free to correct me