The difference between the AGILE and SCRUM methodologies. I am not too clear. Can someone help illustrate in practice?

A scrum team will work to deliver a potentially shippable product increment in each iteration. At the end of each iteration, they will review that increment with the customer and get feedback before planning the next iteration.

When that product gets released is up to the customer or some designated stakeholder (perhaps the Product Owner). This doesn't make the project a "waterfall." They're just managing releases.

On the project on which I worked the customer had both a project director and a project manager and they planned the project as any other waterfall project. The business was involved only at the beginning when the requirements were defined and then at the end during user acceptance testing.

Despite this the software company still used Scrum but nobody from the customer reviewed what was "delivered" after each sprint. The customer didn't care about the sprints at all. Scrum was just used internally by the software development company to plan their work with no input from the customer.

At that time another project was in progress that used Agile. After a while the business said that they can't commit resources to be involved in the planning and reviewing the iterations so the project ended up a fiasco. When the business backed up the team did not have waterfall like requirements to continue the work. The project ended up costing huge sums of money and delivered very little.

The waterfall project also ran over budget but in the end delivered and was very well received by the business. It would have been even better if the software company wouldn't have used Scrum internally for development.