int is a special function prefix that is inserted in a template when trying to template a system message. It is used to replace $foo with {{{foo}}}.

$foo is used for variables. Normally, for a template, {{{foo}}} is used as a variable, however, in the MediaWiki namespace this is not the case.

Compare the difference:

{{MediaWiki:Youhavenewmessages|variable1|variable2}} makes

You have $1 ($2).

However, replace 'MediaWiki' with 'INT' making {{int:Youhavenewmessages|variable1|variable2}} makes

You have variable1 (variable2).