Catalog and catalogue are the same word with the same meaning; catalog is simply the American spelling of catalogue. If you're writing for a British or general audience, use catalogue; American style guides prefer catalog. Either works as a noun (a list of items) or a verb (to make such a list).
The core difference
Catalog is the US spelling; catalogue is the standard spelling everywhere else, including the UK.
- catalog — US spelling: She flicked through the furniture catalog looking for a new sofa.
- catalogue — standard/UK spelling: The library catalogue lets you search by author or title.
- catalogue (verb) — to list or record items: The museum spent months cataloguing the newly discovered artefacts.
How to tell them apart
There's no meaning difference to remember here — only a spelling choice. Catalogue traces back through Old French to Late Latin catalogus and ultimately Greek katalogos, meaning a list or register; catalog is simply that word trimmed to fit American spelling conventions, the same pattern you see in words like dialogue/dialog.
The trick for keeping them straight is regional, not grammatical: British, Irish, Australian, and most international publications use catalogue, while American publications and brands (think mail-order catalogs) use catalog. Pick the spelling that matches your audience and stay consistent throughout a document — mixing the two in the same piece of writing looks like a typo rather than a style choice.