PaaS and Serverless are certainly closely related. The main differences so far seem to be:
Unit of execution: with PaaS you’re dealing with a set of functions or methods, in Serverless land with single functions.
Complexity: with PaaS you have to conform to a number of (contextual) requirements, need to set up stuff, etc. while with Serverless you only need to specify your function.
Pricing: with PaaS you’re paying for the whole package and in Serverless land only per (successful) function call/execution.