Cost - Our primary data store has >1 Petabyte of raw data stored across dozens of Postgres instances. The amount of data we store is at the point where RDS is too expensive for us. The cost of an instance on RDS is more than twice the cost on EC2. For example, an on-demand r4.8xl on EC2 instance costs $2.13 an hour, while an RDS r4.8xl costs $4.80 an hour.
Performance - The only kind of disk available on RDS is EBS. EBS is slow compared to the NVMe the i3s provide. We used to use r3s with EBS and got a major speedup when we switched to i3s. As a side note, the cost of an i3 is also less than the cost of an r3 with an equivalent amount of EBS.
Configuration - By using EC2 we can configure our machines in ways we wouldn't be able to if we used RDS. For example, we run ZFS on our EC2 instances which compresses our data by 2x. By compressing our data, we get a major cost saving and a major performance boost at the same time! There isn't an easy way to compress your data if you use RDS.
Introspection - There are times where we've needed to debug performance problems with Postgres and EXPLAIN ANALYZE won't suffice. A good example is we used flame graphs to see what Postgres was using CPU for. We made a small change that resulted in a 10x improvement to ingestion throughput. If you are curious, I wrote a blog post on this investigation: https://heapanalytics.com/blog/engineering/basic-performance...