AddType principally adds a content type header to the HTML page that is output.
AddHandler principally tells Apache to invoke a PHP handler for .php files. There are several ways of doing that, hence several variations in the syntax.
The functionality and the syntax has also varied between Apache 1.3, 2.2 and 2.4.
As I can never remember the "right way" for each version, and what you need to do can also depend on other server settings set by the host, it often is a case of trying different things until it works.