Oh boy, I'm an engineer, but I'll do my best to pretend I'm on the sales or marketing team for a minute...
First of all, CockroachDB Serverless is available on AWS, and should integrate quite well with that ecosystem, including with Serverless functions offered by AWS Lambda.
Here are a few advantages of CockroachDB Serverless that Aurora will struggle to match (note that we're still working on Serverless multi-region support):
1. Free-forever tier. We offer a generous "free forever" tier that doesn't end after a month or a year. As the blog post outlines, our architecture is custom-built to make this economical.
2. No ceiling on write scalability. Even non-Serverless Aurora runs into increasing trouble as the number of writes / second increases past what a single machine can handle. CockroachDB just keeps going. We've had multiple high-scale customers who hit Aurora limits and had to move over to Cockroach to support business growth.
3. True multi-region support. Aurora only allows read-only, stale replicas in other regions, while CRDB allows full ACID SQL transactions. If you want to move into other regions of the world and have latency concerns or GDPR concerns, CRDB is custom-built to make the full SQL experience possible.
4. No Cloud lock-in. Perhaps this is not a concern for you company, but many companies don't like getting completely locked in to a single Cloud provider. CockroachDB works on multiple cloud providers and doesn't have a monetary interest in locking you in to just one.
5. Online schema changes. CockroachDB supports operations like adding/removing columns, renaming tables, and adding constraints without any downtime. You can perform arbitrary schema changes without disturbing your running application workloads. SQL DDL "just works".
6. Cold start in an instant. CockroachDB clusters automatically "scale to zero" when they're not in use. When traffic arrives, they resume in a fraction of a second. Compare that to Aurora, where you need to either have a minimum compute reservation, or you need to endure multi-second cold starts.
7. Great support. We've got a friendly Slack room where you can get free support and rub shoulders with fellow CockroachDB users, as well as CockroachDB folks like myself. We also have 24/7 paid support for deeper problems you might encounter.
Taken altogether, CockroachDB can go wherever your business needs it to go, without all the constraints that traditional SQL databases usually have. Do you want thousands of clusters for testing/development/tiny apps at a reasonable cost? Could your business take off and need the scale that CRDB offers? Could your business need to expand into multiple geographic regions? Are some of your workloads erratic or periodic, but still should start up instantly when needed? It's not just about what you need now, but what you may need in the future. It makes sense to plan ahead and go with a database that has "got you covered" wherever you need to go.