What is the difference between catalogue and catalog? I cannot really decide which one to use for a product catalogue for a shop.
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.