RDS

Relational DB Service with backups and scaling (vertical/horizontal)
Key thing is that it does not appear as a DBaaS but as Database Server as a Service (DBSaaS)
Multiple databases can be setup on one DB server. There is not SSH/OS access.

Multi-AZ setup for ONLY DR. Amazon RDS Multi AZ Deployments | Cloud Relational Database | Amazon Web Services
Amazon Aurora is not RDS, but has similarities and integrates with RDS.

RDS Proxy

It maintains a pools of connections which are open and closing, but your applications only access the proxy. Your application won’t need to establish new TCP/UDP connections to a DB. Fully managed (for you), has autoscaling, High availability (HA) by default, only accessible from VPC, access vi proxy endpoing, can enforce TLS and SSL, can reduce failover time by over 60% for Amazon Aurora, abstracts failure for application (application will just wait until Proxy makes a connection to the other DB in case of DB fail)

On can help to reduce number of direct connections to DB by using Multiplexing - Wikipedia
Useful when:

• Too many connections errors…
• DB Instances using T2/T3 (i.e smaller/burst) instances
• AWS Lambda … time saved/connection reuse & IAM Auth
• Long running connections (SAAS apps) - low latency
• Where resilience to database failure is a priority…
• … RDS proxy can reduce the time for failover
• .. and make it transparent to the application

Billing

Billed per Instance size/type (like in EC2) – hourly per second
Multi-AZ extra additional costs
Storage type & amount (like EBS)
Data transfer
Backups & Snapshots (you receive free storage for backup which equal to your storage disk size)
GB per month
Licensing, commercial plans