"Catalogue" versus "catalog"

They have the same meaning. Catalogue is used in British English whereas catalog is mostly used in American English. I've also seen cases where catalogue was used by some American friends, though.

Although both spellings originally referred to the same idea, it's my personal experience--in the U.S., at least--that these two forms have diverged over time to encompass separate domains based on the locus of ideas evoked by the word in question.

Example 1: "Dialogue" refers to an undefined, dynamic exchange between two active parties (e.g., dialogue between two characters) whereas "dialog" will typically refer to a one-sided exchange--as with a computer prompt with a deterministic output (e.g., dialog box).

Example 2: "Analog" (adj) almost exclusively refers to "non-digital" technology (e.g., analog amplifier), while "analogue" (n) is used almost exclusively to describe things in figurative/metaphorical domains.

Example 3: "Catalog" and "catalogue" are a good example of this. Catalog often seems relegated to static, physical objects (e.g., the Sears catalog) while catalogue may refer to a more abstract (non-physical) concept of grouped elements (e.g., the Beatles' catalogue of music).

In all three cases, both spellings are used heavily in American English, to different effect; I cannot speak to British (or other regional) English.