Can anyone explain the key difference between key-user extensibility and side-by-side extensibility in SAP?
In-App or Key-User extensibility are means provided by the SAP product that enables you to extend the apps. For example one may add a new custom field to the UI which is then saved together with the business object instance. It can even be forwarded to follow-up documents in case of process extensibility. Further options of key-user extensibility are own business objects or business logic modifications in pre-defined extension points. This concept applies for cloud software. OnPremise customers have other options to get such kind of enhancements because they have direct access to the code.
Side-by-Side extensibility refers to enhancements that you create on a different platform (e.g. SAP Cloud Platform) but integrate with your SAP product. As an example, you might react on a certain event in your SAP product, notify your own SCP implementation to handle accordingly. Or you have an own app on SCP that writes data back into the SAP product. The communication works via public APIs in such cases.
Here is the reference from SAP which is quite clear end self-explanatory: Extensibility Explorer