Adobe After Effects can render its own projects (.aep) as MPEG-4 files (.mp4). This can be done using the program's default Composition ► Pre-render... function. This is used to add visual effects to multimedia files and is likely the only way to convert aep to mp4.
As far as we know, this is the only way how to perform this particular aep to mp4 conversion and no 3rd party aep converter exists.
Updated: April 19, 2017