Since Gradle uses Groovy instead of XML for it's configuration scripts, it achieves a great deal of flexibility but unfortunately that comes with slower builds when the project gets large enough. This is because the entire Groovy script has to be recompiled for every build instead of simply being parsed.
Android Studio's build system is an Android plugin for Gradle. What's more is that the Android Gradle plugin can be installed and run even on machines that don't have Android Studio, which enables you to build Android apps everywhere (for example continuous integration servers).
Gradle is a dependency programming tool first and foremost. Gradle will make sure that all declared dependencies are properly executed for every random task that you execute in your setup. The code can be spread across many directories in any kind of file layout.
Since Gradle does not use XML but it uses it's own DSL based on Groovy, Gradle scripts tend to be shorter than other build tools that use XML. Boilerplate code is also considerably small because it's DSL is designed to solve a specific problem: moving the software through its lifecycle starting from compilation into static analysis and testing, packaging and finally deployment.
Gradle has full integration with Jetbrains IDEA.
IDEA understands multi-module Gradle builds and automatically maintains the IDEA modules within the project.
You also have the option to run unit tests with either the built-in JUnit/TestNG test runner, or delegate running the test to Gradle using the same visualization as the built-in runner.
Gradle integrates perfectly with Ant. Giving developers a lot of Ant's flexibility. You can run individual Ant tasks on Gradle or even entire Ant builds.
Gradle follows the convention over configuration paradigm in order to make it easier for developers by having already made a number of decisions out of the box.
Maven is very opinionated and declarative. This means that most developers will easily understand how your build system works. They don't have to guess which file will end up in the jar. An IDE can also import any project and easily reason how everything is organized and fits.
Apache Maven is very popular in the Open Source community. With many open source projects using it as their build tool of choice. Considering that the Open Source community is rather large and usually helpful so there are a lot of guides and tutorials out there written by third-party sources on Maven.
While it's true that Ant gives you a lot of freedom and control over your builds, it also means that you will have to constantly reinvent the wheel on how your project structure and how everything fits together for every new project. This also makes it harder for other people to simply start working on your project because of the wide range of different ways to build a project with Ant. While tools that have more constraints generally end up looking the same and it's easier to understand how everything fits.
Ant is an imperative build system, this means that it's very good at controlling the build process. You "tell" Ant what to do during the whole process, for example: "compile these files and put them in that folder". This is great for special projects where you want as much control as possible during the build process.
Ivy is a powerful tool for managing dependencies. It's highly configurable and very flexible and is not tied to any specific structure or architecture.
Although it can also be used as a standalone tool, it's tightly integrated with Ant and provides all the dependency management services for it, basically transferring all of Ivy's flexibility in dependency management to Ant as well.