Project Online would be the equivalent of a local deployment of project server. In general online is constantly updated and hosted by Microsoft where a Local Project Server deployment is host in your data center and updated are deployed by your administrator. My understanding is Project Server/ Project Online are consider feature complete.
Project Web is based on the Microsoft Dynamics and Power app platform and is the future however it is not currently feature complete to the level of Project Server / Online. Meaning Project Server / Online is still more feature rich than project web currently. I haven’t seen any final decision on any end of life and the consultants I have talk with are anticipate beyond 2025 because no formal migration path has been identified. Look at the video and watch the comparison and see what works for your organization. I hope this points you in a good direction, I am currently using project online and web as a part of my Project P3 subscription and looking at PPM solutions using Microsoft Dynamic/Power BI/Power App solutions from a couple of vendors.