The agile approach to project management does not necessarily mean savings in terms of either time or costs.
The very idea that the requirements can be continually redefined contains within itself the possibility of infinite processing cycles and possible reworking with the consequent unpredictability of the completion times of a project.
Agile is in fashion. It's a fascinating concept that projects can be managed without the need to apply cascading project management principles and techniques. In particular, the absence of structured project documentation as a principle in the agile approach is in my opinion one of the weaknesses of this approach. As well as the lack of a project manager who takes full responsibility for the project.
Perhaps the main benefit of the agile approach is that with Agile an application can also evolve during the development phase and therefore be more responsive to the client's expectations.
"The real value in agile is in fact in its flexibility and responsiveness to evolving business requirements, which ensures that the organization gets a system that genuinely addresses its needs, rather than the system it thought it wanted at the beginning of the process (as would be the case in a waterfall project, where full requirements are defined at the start and remain relatively static). The reality is that business requirements evolve over time, and agile makes it easier for projects to adapt accordingly. "
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The other advantage is that constitutes a series of common sense principles that can guide the work of a project team even in the absence of a project manager or a project management methodology shared at company level.