The data 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 data downloader is found within the record editor for a specific app. This option will provide you with an export file that contains your filtered records with the option to include any associated photos, signatures, and audio files. If you are looking to exclude certain columns or wish to filter out a subset of records, then you will want to use this download 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.
Clicking the "Download Data" button presents you with the download options dialog, which allows you to select your data format and if you have media and/or record link fields, you can specify whether the values written should be logical (media file URL, linked record title) or relational (file/record ID). If you need to join the the records and photos, you'll want to select the resource ID, otherwise, you'll probably want to use the logical options.
Note: Download files are currently limited to 10GB in file size and/or 100,000 media files.
The Recent Downloads picklist contains up to 20 of your most recent downloads. Downloads expire after 7 days and will be removed from the list, but you can always generate another download if necessary.