At a high-level they're very similar in the problem they solve, both are focused on giving you reliable disaster recovery. At the time Wal-E was written barman didn't exist and S3 was one of the few reliable options to backup to (this was over 5 years ago). Since then Wal-E has expanded to include just about every object store you could want, and at the same time 2Q introduced barman as their take on it.
Wal-E has been used for a number of years to provide disaster recovery for Heroku Postgres, for over a million databases. It enables their follow and fork functionality, and we're using it at Citus as well for Citus Cloud given we have the person that authored it.
As for exact differences I'm less familiar as I've not seriously run barman in production so perhaps someone that's run both can chime in.