Pig allows one to load data and user code at any point in the pipeline. This is can be particularly important if the data is a streaming data, for example data from satellites or instruments.
Hive, which is RDBMS based, needs the data to be first imported (or loaded) and after that it can be worked upon. So if you were using Hive on streaming data, you would have to keep filling buckets (or files) and use hive on each filled bucket, while using other buckets to keep storing the newly arriving data.
Pig also uses lazy evaluation. It allows greater ease of programming and one can use it to analyze data in different ways with more freedom than in an SQL like language like Hive. So if you really wanted to analyze matrices or patterns in some unstructured data you had, and wanted to do interesting calculations on them, with Pig you can go some fair distance, while with Hive, you need something else to play with the results.
Pig is faster in the data import but slower in actual execution than an RDBMS friendly language like Hive.
Pig is well suited to parallelization and so it possibly has an edge for systems where the datasets are huge, i.e. in systems where you are concerned more about the throughput of your results than the latency (the time to get any particular datum of result).