Anupam, I agree with Sergio. There should not be any at a fundamental level. if I has stakeholders who always worked in a waterfall set up and are moving to an agile set up, I will explain the difference between the two in a language they understand. Saving Changes...
I agree with Sergio and Niraj, but to me is a "yes and no" answer.
There's always a big dependency on the type of Organization/Business your are working at.
In a pure agile environment (say Scrum for the sake of the example.) is the Product Owner (with guidance from the Scrum Master) the one responsible for managing the Stakeholders, and it's up him/her to do so in the best possible way.
With waterfall as PM you can find defined guidelines and tools to manage stakeholders and the way it works for me is leveraging that knowledge and adapting it as part of the agile practice.
What I mean with this is:
In a waterfall organization you will have as part of your meeting plan, stackeholder check-ins, demos, feedback session, etc. You will have identified risks tied to stakeholders expectations, their weight within the organization, etc.
In an agile organization you will have a similar approach, but the way you will manage that could be slightly different. And what is different is not the what but the how/when.
You will probably engage stakeholders more frequently, e.g. after every iteration/sprint for a sprint demo and feedback. You will do so in a more agile way.
The problem you will address is the same, some tools will be the same but the approach will be different. Otherwise agile would not be agile and waterfall would not be waterfall.
Rodrigo brings up an important difference. The Product Owner (PO) is responsible for working with the stakeholders, as opposed to the project manager. Part of the job of the Scrum Master is to make sure that the PO is working with the stakeholders to update and prioritize the backlog and roadmap.
It's easy to say that there is no difference, but ask a PO who has not received training based on the PMBOK, and the PO may not know what you're talking about. The Scrum Master also may not have received such training. Neither CSM nor CSPO training goes in depth into stakeholder management. Stakeholder management may be the same for the experienced professional, and in theory, it's easy to say that stakeholder management is stakeholder management regardless of the project approach, but if you have a formal stakeholder management process on a Scrum project, you are most likely borrowing from another approach. Saving Changes...
Mr. Porter brings up a good point about responsibility in stakeholder management. The Project Owner/Sponsor has equal responsibility for managing the deliverables of stakeholders whether in an Agile or Waterfall environment. Saving Changes...
In general, stakeholder engagement is not affected by the lifecycle used but who is on "first" for specific stakeholder interactions and where/how stakeholder info is tracked may. For example, a PO might be responsible for being first point of contact for certain types of stakeholders. Saving Changes...