I think the biggest difference is the scalability built-in. It has a beautiful Web admin and CLI admin that ease the scailability works including sharding and adding a node to the cluster.
PostgreSQL also has built-in streaming replication, which is useful and works nicely. But the scenario it covers is scaling reads, not writes. To scale writes, you should shard the data manually.
There are efforts to bring write scalability to PostgreSQL. One of them I'm keeping my eyes on recently is Postgres-XL, which uses statement-based replication that focus on writability and availability. It is a premature product that started recently, though. And also, some easy-to-administer features like automatic failover and configuration propagation (adding/removing nodes) are not the objective of the project. You should adopt another solution, or write it manually.
However, instead of getting the scalability, you lose ACID in RethinkDB - whose impact can be huge on some use cases. But personally I think it is worth losing to gain scalability easily. And there is two-phase commit which is at least doable to imitate transactions, though it isn't elegant at all.
If your application is read-intensive, I think staring with PostgreSQL and adopting Postgres-XL or pgpool later is a good strategy. But if you are to shard the data to scale writes, or you want an all-on-one solution that helps the administrative works a lot, I think RethinkDB is highly recommended.
Postgres is an amazing product, and incredibly stable by virtue of being very mature. You can't go wrong with it and will almost certainly be just fine.
However, you can't quite compare Postgres document support with RethinkDB. You'd really need to play with the product to understand the difference, but think of it this way. You could sort of treat Java as a dynamic language by casting everything to object, it would work just fine, but it is night and day compared to how it would work in Python. The difference between the experience of using Postgres and RethinkDB for JSON data is just as vast. You'd really how to try both before you can viscerally understand how amazing a dedicated JSON environment can be.
RethinkDB is behind Postgres on raw performance and we're still shaking out scalability quirks (see http://rethinkdb.com/stability/) but if you take the long view (a year or so), these will be worked out in on time. I'd encourage you to play with the environment -- it's easy and fun, and see if you like it. Feel free to shoot me an email (slava@rethinkdb.com) and I'll be happy to help you out if you have questions.
RethinkDB looks better than MongoDB in many respects. But how does it compare to Postgresql (latest versions with document support)?
If I were to start on a new project with modest scalability requirements, what should I choose and why?