What is the difference between Agile and Scrum project methodologies?

The comparison is not correct. Agile is not a methodology, it's a framework!

Agile is a set of abstract rules for which you need a well defined methodology to implement.

We have two main application development frameworks: Waterfall and Agile.

A framework gives you a set of rules to stick when you want to develop a software but it does not tell you how! Methodologies tell you HOW to act under a specific framework. for instance scrum tells you how to perform agile!

Waterfall is the old sequential framework and it has its disadvantages specially for big projects and there are lots of methodologies designed to do it. For agile, the most common methodology is scrum.

Both Agile and Waterfall have their own advantages and disadvantages, but agile seems to be the one that covers most cases, and specially works best where waterfall fails.

That's it.

*In simple way Agile is an software development life cycle similar to SDLC.*Scrum is one of the wildly used flavor of Agile.

eg: SDLC having flovor like - Waterfall, RAD, V-model, Spiral  .   in same way  Agile having Extreme Programming(XP) ,Adaptive Software Development (ASD),

• Dynamic Systems Development Method (DSDM) ,Scrum ,Crystal ,Feature Driven Development (FDD).

*Scrum is mostly benefitial for product or services which are having chances of lot of change request, as well if project does not have much clarity on primary phase even though we can start project with scrum methodology.Scrum welcome to changes as its base is inspect & adapt, by keeping small duration of incremental process, which fasten the process and provide maximum customer satisfactory output.