MODification vs Extension
What is a MODification?
A modification is a change in the core files of phpBB, hence the term. The core files are modified in order to change their function. This is basically the guttural way to add or change functionality in phpBB in the modern era since editing the core code can cause problems. It was however the only way to add features to phpBB2 and phpBB 3.0.x, since the plugin architecture for Extensions did not exist at that time, and thus, they are the predicessor to modern day extensions. Since phpBB is open source however, MODifications have never truly gone away. While they are not placed in official databases, they are de facto still written and can be found in phpBB Custom Coding on phpBB.com as well as a few being listed on this very site. For simple changes (and mainly simple changes) where one does not want to plug into the php events for extensions, they are a somewhat viable alternative.
What is an Extension?
An extension adds or changes phpBB functionality WITHOUT altering the core code/files. The only changes file-wise is a few files and folders dropped into the ext directory. This is the more modern way to add functionality to phpBB and it is considered superior as they are easier to disable and remove. They are more convenient for the end user, but may be inconvenient for those who just want to make minor tweaks to their board, depending on what the user is trying to do. That being said, it is better to code an extension than making a MOD, and MODs are liable to be wiped out when the board is updated, unless the Advanced Update option is used. However, the Advanced Update option's days are numbered and it will no doubt be removed by phpBB 3.4.0 or phpBB 4.