The serverless framework is an open source tool that you can use to "deploy" "serverless" assets. Basically it covers most of those scripts you usually have to write and gives you a declarative format with some great variable capabilities that help you managed config and declare your components clearly. Running the tool deploys to your cloud of choice and results in functions, APIs, workflows, databases, or whatever you declare now existing and ready for use.

That framework is separate from the class of services regarded as serverless. The star that kicked it off was AWS Lambda. Serverless colloquially means you are dealing with an abstract service contract rather than a server (e.g. the oldest is S3). This removes patching and other maintenance that usually does not directly support business value. More formally, serverless includes auto-scaling to zero, paying only for what you use, high availability, and other design patterns most outside the large tech houses cannot use at low to no cost.

Your VPS is always on and always paid for. It can crash or get in a bad state. It can go out of date and need patching. It is mutable and more vulnerable. It is limited in resources.

Nike reported AWS Lambda scaling in production at 20K RPS/S (0RPS@0s, 20K RPS@1s, 60K@2s, 120K@3s, ...).

serverless-artillery, with loosened account "safety" limits can scale from nothing to producing billions of requests per second on target systems.

Full disclosure: I contributed to the serverless framework and serverless-artillery. I'm a biased fanboy.