This document is intended for those who are trying to decide which format to use for their feeds. If you're solely an R3R user, this probably doesn't concern you, being that R3R can extract the data from five feed formats. However, you may want to read this if a feed to which you want to subscribe is available in multiple formats.
Note the wording, "extract data." I try to avoid using the term support, because R3R isn't technically specification-supporting. It's designed to be liberal, and only understands a subset of some of the specifications. See Format Support Status for more information.
Below are the advantages and disadvantages of the various formats R3R understands, in more-or-less alphabetical order.
Atom is perhaps the most exciting feed format, as it is an open format standardized by the IETF. This is its advantage.
The disadvantage is the large amount of required metadata that is not needed for many, if not most, feeds; hence, Atom feeds are larger and more difficult to created than RSS.
ESF isn't being standardized, but it is a very simple format that's easy to author and generate, and has the smallest file sizes (read: lightning-fast download).
R3R can extract the information from three different RSS formats
These formats use RDF, 1.0 more so than 0.9. R3R will ignore the RDF and extract the actual data. These are a waste of bandwidth, though RSS 1.0 is as close to an open standard as RSS gets.
These are fairly the "de facto" formats for feeds. They represent a middle ground — not minimal like ESF and not overloaded like Atom. These versions are based on pure XML, instead of an XML serialization of RDF.
RSS 2.0 is designed to be the last XML-based version, with future extensions done via XML namespaces.
RSS 3.0 is most similar to RSS 0.9, except that it uses a MIME serialization of RDF instad of an XML serialization.
RSS 3.0 packs all the advantages of older versions, but is more compact than XML. But note that few take it seriously, so don't expect parsers to be global. For global distribution, RSS 0.91 is probably your best bet.