Checkstyle can be used to check for adherence to specific coding standards/conventions. This does not change your code, but only flags lines that are not compliant.
The Formatter is used to make all source code follow specified formatting conventions. It does change your code, e.g. it may indent a certain amount, move brackets to a new line, break long lines, or other such operations.
Cleanup is more generic and may also change your code. It may call e.g. the formatter, organise imports (adding or removing them as necessary), add missing annotations, or remove unused code.
In a nutshell, Checkstyle only checks for rules, but does not modify code, the Formatter modifies source files but only does changes related to formatting, while Cleanup not only changes source files, but may also change code by adding or removing lines.