Are sweepstakes and prize draws the same thing?
In substance, yes. Both describe a promotion where a prize is awarded to entrants by chance, and the main difference is which side of the world you are on. Sweepstakes is the American term; in Australia the same promotion is usually called a prize draw, and its formal regulatory name is a trade promotion. The mechanic is identical, but the label and the rules attached to it change by country.
Is a contest or a giveaway the same thing?
Not quite, and the difference is how the winner is chosen. Sweepstakes and prize draws pick winners by chance, usually a random draw, while a contest or competition, in the strict sense, is won on merit: the best photo, the cleverest answer, judged rather than drawn. That skill element is not just wording, because in some markets it changes how the promotion is regulated. Giveaway is a loose everyday word that can mean either, so it pays to be clear whether yours is decided by chance or by skill.
Do the rules differ, or just the words?
The rules genuinely differ, and this is where the two stop being interchangeable. In the United States a sweepstakes must be free to enter: requiring a purchase adds what the law calls consideration, and prize plus chance plus consideration is classed as a lottery, which only a government may run, so US promotions always offer a no-purchase way in. In Australia a trade promotion can tie entry to a purchase, but a chance-based one needs approval in some states, for example an authority in New South Wales once total prizes pass $10,000 and a permit in the ACT above $3,000. Same mechanic, different legal frame.
Which term should a brand use?
Match the audience, not your own habit. If your entrants are American, sweepstakes is the word they expect; in Australia, prize draw is clearer to customers and trade promotion is the term regulators use. The word is a marketing choice that costs nothing to get right, so pick the one your entrants already say and keep it consistent from the entry page to the winner announcement. The harder part is running it well, which is what how to run a compliant prize draw is about.
Does the name change how you run it?
No. Whatever you call it, a prize-by-chance promotion has to be set up to the same standard: prizes set and published before entries open, draws that are random and auditable, and the right approvals for each market it reaches. Those are engineering and record-keeping questions, not naming ones. Run it on a platform with certified random draws and audit-ready records and the same promotion can go out as a sweepstakes in one country and a prize draw in another.