What are some key difference in functionality between ProM and Apromore?
Below are three key differences.

Apromore is a server-side collaborative tool. It is typically installed on a server (on-premise or on the cloud). Users connect to Apromore via the Web browser. Users can upload process models and event logs, organize them into folders, enrich them with log filters and dashboard designs, and share them with other users. Apromore comes with a process model editor, so you can edit the process models you discover from your event logs, beautify them, and export them in BPMN format (so you can share them with other tools). Meanwhile, ProM is a desktop tool. You typically install it in your laptop. You use it to analyze an event log locally, but you cannot use it in a collaborative manner with say 3-4 team members. You cannot save your analysis (e.g. custom dashboards, filters) in ProM.
A second difference is that Apromore supports BPMN natively. The process discovery algorithm in Apromore is Split Miner, which discovers BPMN models by design. Traditionally, ProM is designed to support an academic notation called Petri nets, which is used in research studies but generally not used by process improvement practitioners.
Finally, Apromore supports both process mining use cases (discovery, conformance), but also process performance monitoring use cases via customizable process performance dashboards.