The biggest limits are based on the fact that Aurora Postgres doesn't use a filesystem. Rather, Amazon has created a block-based back end for Postgres that allows clustered access to the data store. The data store itself, like EBS, is replicated for performance and redundancy. This has some interesting side effects. Postgres was built around the assumption that the filesystem cache was the primary block cache. You allocate a fairly limited amount of memory to the internal Postgres shared memory pool and leave the rest to be used by the filesystem block cache. Aurora Postgres, on the other hand, must assign that memory to the internal Postgres shared memory pool in order to serve as cache since there is no filesystem and thus no filesystem block cache. Unlike the filesystem block cache pool, Postgres jobs cannot take memory away from the internal shared memory pool in order to accomplish whatever task they are doing. The end result is that internal jobs that require a lot of memory can die with out of memory errors since there's not enough memory outside the Postgres shared memory pool to allocate for that job.
The other big limitation is that Aurora Postgres has limited space for handling large sorts or indexing operations. Regular Postgres uses a directory, pgsql_tmp, in a tablespace to store temporary heap results for sorts and indexes too big to fit in work_mem (which by default is 2gb). This can be as big as your filesystem allows. If, for example, I have 500gb free in my tablespace, I have no trouble sorting an entire 150gb table into an arbitrary order then exporting it to an external consumer.
But remember, Aurora Postgres doesn't have a filesystem for its tablespace. It has a block store. Instead, Aurora Postgres instances that are doing large sorts or indexing large files use local storage, which is currently 2x the size of memory. That is, if an Aurora database instance has 72gb of memory, you only have 144gb of temporary space. Good luck sorting that 150gb table.
What this means for me is that Aurora Postgres has some interesting scalability limits when dealing with very large data sets. I'm currently managing about 2 billion rows in Postgres. Needless to say, this requires a lot of very large indexes in order to segment this data space into usable consumable subsets. Creating these indexes is a slow and tedious problem on Aurora Postgres because you have to do them one at a time, you can't do them in parallel, due to the lack of temporary space to use for the sort heaps. And if I'm querying and sorting significant subsets of this database, again Aurora Postgres has some serious limits due to the inability to expand pgsql_tmp.
My guess is that people who are dealing with much fewer rows, but who are querying those rows with much greater frequency, will have a more successful experience with Aurora Postgres. But then they'll run into the IOPS costs. Basically, to get the same IOPS that would cost me $1200/month on EBS, I'd end up paying around $4,000/month on Aurora.
So: What's the point of Aurora? Aurora does have a couple of positives. You can create additional read replicas virtually instantly, since they're just pointed at the same shared block storage. Failover simply happens, and happens almost instantly. And from a management point of view, Aurora makes the database administrator's job far simpler since you no longer have to closely monitor your tablespaces and expand your block storage as needed (and reallocate tables and indexes across multiple tablespaces using pg_repack) in order to handle growing your dataset. Still, in its current state of development, given its limitations and high costs compared to running your own cluster, I really cannot recommend Aurora Postgres.