Explanation:
Predictive:
Requirements: are controlled as requirements and constraints emerge.
Changes: are elaborated frequently during delivery.
Risk and Cost: are continuously involved.
Key Stakeholders: are controlled by detailed planning.
Agile:
Requirements: are defined up-front before development team begins.
Changes: are constrained as much as possible.
Risk and Cost: are incorporated in real-time during delivery.
Key Stakeholders: are involved at specific milestones.
The predictive project approach, often referred to as the waterfall model, is characterized by its sequential phase-to-phase development where requirements and planning are done up-front. In contrast, the agile approach is iterative and incremental, with requirements and solutions evolving through collaboration between self-organizing cross-functional teams.
In the predictive approach:
Requirements are detailed early in the project and changes are managed through a formal process.
Changes are not expected to be frequent and are carefully managed.
Risk and Cost are analyzed and assessed as part of the initial planning.
Key Stakeholders are identified and their needs are addressed through detailed planning.
In the agile approach:
Requirements are often co-created with stakeholders and can evolve as the project progresses.
Changes are expected and embraced as part of the methodology to adapt to evolving needs.
Risk and Cost are managed in real-time, allowing for quick responses to change.
Key Stakeholders are engaged continuously, with their feedback incorporated into the project iteratively.
The PMBOK Guide1 and the Agile Practice Guide provide a framework for understanding these approaches. The PMP Examination Content Outline2 also details the differences between predictive and agile methodologies, emphasizing the importance of choosing the right approach based on the project's needs and environment.

The project manager should educate the new sponsor about the agile approach for project delivery, which is an iterative and adaptive way of working that focuses on delivering value to the customer and responding to change. The project manager should also explain how agile teams are self-organized, meaning that they have the autonomy and authority to plan, execute, and monitor their own work, without excessive control or interference from the sponsor or other stakeholders. The project manager should emphasize the benefits of self-organization, such as increased motivation, collaboration, innovation, and quality. The project manager should also clarify the role and responsibilities of the sponsor in agile projects, which include providing vision, funding, support, and feedback, but not micromanaging or dictating the work of the team. 123 References:
* 1: What is agile project management? - APM
* 2: What is Agile? - Project Management Institute
* 3: Role of Project Sponsor in Agile Project Management - PM Majik

In a scenario where a company is not yet profitable and is facing project delays with an overwhelmed staff, the project manager should prioritize the projects based on their potential to increase profit. This is in line with the principles of project financial management, which suggest that prioritizing projects that align with the company's strategic goals, such as financial objectives, is crucial. By focusing on projects that are more likely to contribute to the company's profitability, the project manager can help ensure that resources are allocated effectively and that the company's financial position is strengthened. This approach is supported by the PMBOK Guide, which emphasizes the importance of aligning projects with organizational strategy and business objectives.
References:
PMBOK Guide - Sixth Edition
PMP Examination Content Outline - June 20191
Professional in Business Analysis Reference Materials2
Project Prioritization Process: 5 Strategies to Prioritize Your Projects3 Analytic Hierarchy Process Prioritize Projects | PMI4

When a project manager returns from an absence to find a nearly complete project schedule developed by the team, the priority should be to ensure that the schedule aligns with the project's vision and objectives. This is crucial because the project scope statement, which includes the project's vision, objectives, and work breakdown structure, guides all project activities. The project manager must review the developed schedule against the approved project scope statement to ensure that all project work aligns with the documented objectives and deliverables. If discrepancies are found, the project manager should work with the team to make necessary adjustments. This approach helps maintain the project's integrity and ensures that the final deliverables will meet the stakeholders' expectations.
(Based on the Project Management Professional (PMP) guidelines and standards from PMI's resources123