Every December somebody proposes a White Elephant and somebody else hears “Secret Santa”, and the confusion holds until the party, when half the room turns up with a gift for a specific person.

They are different games. Not variants — different games, with a different point.

What each one actually is

Secret Santa. Names are drawn in advance. You know who you are buying for, and they do not know it is you. The gift is chosen for a particular person and opened by them.

White Elephant — also Yankee Swap, Dirty Santa, or a dozen local names. Nobody is assigned to anybody. Everyone brings one wrapped gift for the pile, draws a number, and takes turns either opening a new gift or stealing one already opened. The fun is in the stealing.

The difference is not the wrapping. It is who the gift is for: a person, or the pile.

Which one suits which group

A group that knows each other well: Secret Santa. The whole value is in a gift that shows somebody paid attention, and that only works when you know who you have.

A group that mostly does not: White Elephant. Twenty people from four departments cannot buy for each other meaningfully, and asking them to try produces twenty candles. A pile of neutral gifts and a stealing rule is the better evening.

A group that has done the same thing five years running: either, with a constraint. The staleness usually comes from the brief, not the format — twelve ways to change it without changing the mechanism.

A group where money is uneven: Secret Santa, low ceiling. White Elephant makes the cheapest gift visible to everyone, in public, in a game where nobody steals it.

A group that wants to be silly: White Elephant, or a worst-gift contest. The second is the version where being terrible is the actual goal, with rules of its own.

The rules White Elephant needs

It looks like it has no rules and it falls apart without three.

A steal limit. Two or three per gift, then it is frozen. Without this, one popular gift moves eleven times and the game stops being about anything else.
No stealing back immediately. Whoever has just been robbed cannot take that gift straight back. This is the single rule that stops two people from playing tennis for ten minutes.
The first player goes again. Person one drew before anything was open, so give them a final turn at the end. Otherwise the draw order is simply unfair.
One ceiling, said out loud. Twenty or twenty-five, and no wildly better gift smuggled in — that is the one that unbalances the whole pile.

The rules Secret Santa needs

Fewer, and all of them decided before the draw rather than during it.

A budget as a range, not a ceiling. A single number quietly excludes somebody.
Exclusions. Couples and people who share a home should not draw each other; they will shop together and there is no surprise left to protect.
A date, and where the gifts get opened.
Something like a brief. “Twenty euros” produces mediocre gifts; “twenty euros, something edible or drinkable” produces good ones.

Doing both

Some groups run a Secret Santa for the team and a White Elephant for the wider department party. That works, as long as the two are named clearly and nobody is in one thinking they are in the other.

What does not work is deciding halfway. A pile assembled by people who thought they were buying for a named person will contain four gifts that are obviously for one individual, and stealing those is uncomfortable for everyone.

Where a draw fits

Pindao runs the Secret Santa half: names are drawn, nobody draws themselves, exclusions hold, and the organiser never sees the result either.

For a White Elephant there is nothing to draw — no assignments exist. If you need turn order, numbered slips in a bowl are genuinely the right tool, and this is one of the few places where the paper method has no downside at all: nobody can draw themselves, because nobody is drawing a person.

White Elephant or Secret Santa: which one your group actually wants