Reading the PMBOK I noticed that they almost never use the term Sub Project. Using the PMBOK terminology is a project phase the same as a sub project?
In pmBOK, a subproject is a smaller portion of the overall project created when a project is subdivided into more manageable components or pieces (ref: pmBOK v5, p 564).
A project phase may be a subproject if you decide to divide your project in subprojects to reduce its complexity during execution or to share it between several project managers.
Subprojects are parts of main projects goals. Subprojects Have deliverables to deliver by the end as final product which is part of the main project deliverables. The reason we have subproject to have better control over the main project. Phases have specific milestones and deliverables For specific goals. They show progress of phase. Phases are for project management team so if the performed work is too much this means the phase life cycle would be bigger.
All phases are sub projects but not all sub projects are phases. Phases suggest a linear progression, sub projects that are sequenced in a finish-start relationship, generally speaking. But other types of work that can be labeled a sub project may not fit that sequenced approach. And I think it that difference when you would use the term phase versus sub project despite they share characteristics.