What is the benefit of Distributed cache? Why can we just have the file in HDFS and have the application read it?

This is because distributed cache is much faster. It copies the file to all trackers at the start of the job. Now if the task tracker runs 10 or 100 Mappers or Reducer, it will use the same copy of distributed cache. On the other hand, if you put code in file to read it from HDFS in the MR Job then every Mapper will try to access it from HDFS hence if a TaskTracker run 100 map jobs then it will try to read this file 100 times from HDFS. Also HDFS is not very efficient when used like this.