I have seen that if I right click on a project in Eclipse and choose to run it on a server, then I can see output which means the project is running.
If everything is working fine without Maven, what's the point of using it. How is it different than simply running it via eclipse?
Maven is a build tool (build manager, in fact), similar to ANT. The main job of any build tool is configure the project, compile using required projects and do the final packaging. A build script in your project gives a blue-print of project's deliverable structure. This frees you from any configurable dependencies on specific IDE like Eclipse. All you need to know is the standard command to perform the build and you can build your code almost anywhere.
Now, back to your question, why wouldn't do it in Eclipse?
For a simple project and small team Maven is an overkill. You can easily communicate the configuration, IDE to use, and instruct any special steps to be taken. In big projects, however, there exits lots of loosely coupled dependencies. To start with, there will be different settings for developer machine build, test build and production build. There are requirements to run automated test, integration tests, store the build package (artifact) to a commonly accessible repository, update versions of various modules.
Obviously, if all the steps mentioned above is done manually there are chances of missing a step. Moreover, the manual process is time consuming.
Ideally, you should prefer a tool which fits the best for you. If you think that you're able to achieve what you required without Maven, it makes sense to not to use Maven/build-tool just because everyone uses it.
It is suggested to study automated deployment, this will give you bigger picture on what all the stuffs that you can do with build tools. And if you do not feel that it adds any value to your current process, you probably don't need Maven or any other build tool right now.
Your question does not make much sense. Do you expect your users to access your application from eclipse? If so that is a very strange set up in my opinion.
Perhaps your question should be about how to build your project. Maven provides you a way to centralize dependency libraries across the enterprise. It lets you automate your build process (most likely in conjunction with a CI server like hudson, cruise control, etc). It lets you automate your unit testing. Maven makes the packaging of app very easy to do. A developer does not have to follow arcane set of steps to package an application. You add the right plugin and maven takes care of it as part of the build life cycle. All of this magic can happen because of the principle of convention over configuration. There are many more benefits, I just named a few.
Maven is not replacing how you run the app, rather how you package the app, automate that process, and manage the dependencies of your app.
Some links on why someone should use maven:-
