The downloader is the newer of the two export options. The downloader is designed to be much more flexible than the exporter and allow you to produce a data file that is filtered to much more granular levels.
The downloader is found within the record editor for a specific app. This option will provide you with an export file that contains just the filtered data. So if you are looking to not include certain columns or wish to filter what records are included, then you will want to use this export option.
The total number of records that will be contained in the downloaded file is displayed to the right of the column setup icon above the table/map area.