Some aircraft come with a Pilot Operating Handbook and some come with an Aircraft Flight Manual. Why the different name, and is there a difference between them?
Both a POH and an AFM meet the "Operating Limitations" requirement in the ARROW acronym.
The difference between the two is mainly in length and content: an AFM is usually a thinner document, satisfying the requirements of FAR 23.1581 and not much else, while a POH contains these required items plus other information like system diagrams (The contents & format of a POH are standardized in GAMA's Specification 1).
Parts of the POH (like the Limitations section) are FAA-Approved, and serve as the AFM, and both documents are typically associated with a specific airframe (by serial number).
A better explanation might be this:
AFM is a regulatory document (it's contents are prescribed under the section of the regulations the aircraft was certificated under - Part 23, Part 25, etc). The
POH is a GAMA-defined document whose contents meet the regulatory requirements of an AFM, and present other information in a standardized way so that a pilot can go from a Cessna to a Piper to a Mooney to a Socata and browse the book to learn about the airplane they're about to fly with all the information presented the same way no matter who the manufacturer is.
The other two types of documents you may encounter are an "Owner's Manual" (which usually goes along with a thinner AFM & provides some of the information found in the newer-style POH) and a Pilot Information Manual (PIM) which is a "generic" version of the POH which many pilots buy so they can study the procedures without removing the regulatory document from the aircraft.
Chapter 9 of the Pilot's Handbook of Aeronautical Knowledge talks a little about the differences between the two documents (and a whole lot of other flight documents).
all the above answers are wrong. A POH is a 'kind' of AFM. All aircraft must have an approved AFM. Before 1978 this could be anything (owners manual, POH, etc.). After 1978 GAMA (general aviation manufacturers association decided to standardize the AFM around the POH format. The name POH stuck ... but it is still a kind of AFM.
The POH is the official book of rules for that specific serial number airplane. The AFM is the unofficial/generic one for a type of airplane that may or may not match the one it's in. On occasion you'll run into a book labeled "AFM" that is in fact actually the "POH" (often due to an STC requirement).