In summary, WAL-E is simpler program all around that focuses on cloud storage, barman does more around inventories of backups and file-based backups and configuring Postgres. There are integrative downsides to its span. WAL-E also happens to predate Barman.
I'd say the differences between WAL-G and Barman are similar to WAL-E and Barman, which comes up relatively frequently. https://news.ycombinator.com/item?id=13573481
In summary, WAL-E is simpler program all around that focuses on cloud storage, barman does more around inventories of backups and file-based backups and configuring Postgres. There are integrative downsides to its span. WAL-E also happens to predate Barman.