Both MapReduce mode and local mode seem same to the user but the difference is the way they execute.
In MapReduce mode, Pig script is executed on Hadoop cluster. The Pig scripts are converted into MapReduce jobs and then executed on Hadoop cluster (hdfs)
In this mode, Pig script runs on a Single machine without the need of Hadoop cluster or hdfs. Local mode is used for development purpose to see how the script would behave in an actual environment.