There is no much difference between the functionality between MapRed and MapReduce.
The only major difference is that, in MapRed Process, The data is bought towards the code(logic) for data processing and the latest MapReduce process can support the data pull and push operations. I'll leave you a link about MapReduce so that you can understand it better.
The older classes like MultiplTextOutputFormat are not improvised or added to the latest version. The best suggestion is to use the older version in order to get these features which are missing in the latest version.
Also, the old API has been un-deprecated since 0.21. You can find more information about the new API in this link.