Do I earn more BAT for using Beta or Nightly?
No, you don't.
Occasionally there may be features in one or both of these channels that could increase your potential to see ads which would result in a potential increase in BAT.
For example Nightly was the first place where you could choose to receive up to 10 ads per hour, then it came to Beta later, before finally arriving in the Stable channel.
However there is no direct earnings increase simply as a result of using Beta or Nightly.