Folder integration revision refers to the integration revision as it appears in an artifacts folder structure while file integration revision refers to the integration revision as it appears in an artifacts file name.
Consider a number of integration revision paths according to different build tool standards:
Standard Ivy:
org/module/1.0-20111214124053/jars/module/1.0-20111214124053.jar
Non-unique Maven:
groupId/artifactId/1.0-SNAPSHOT/artifactId-1.0-SNAPSHOT.jar
Unique Maven:
groupId/artifactId/1.0-SNAPSHOT/artifactId-1.0-20111412.124253-1.jar
So as you can see, standard Ivy and non-unique Maven have identical file and folder integration revisions (Ivy has 20111214124053 and Maven has SNAPSHOT) while the unique Maven differs (SNAPSHOT and 20111412.124253-1).
In the context of Artifactorys repository layouts, their value is a customizable regular expression which should reflect the expected value of the integration revision. Providing this information helps Artifactory identify the difference between release and integration revision artifacts and extract the information from the path.