Difference between Point Sprites and Billboards

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Can someone tell me the difference between Point Sprites and Billboards in OpenGL? I read a lot about both of them and I'm getting confused more and more about when to use which of them and whether there is actually a difference?

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Wikipedia only knows about Sprites (Billboard redirects there):

In computer graphics, a sprite is a two-dimensional bitmap that is integrated into a larger scene, most often in a 2D video game. Originally, the term sprite referred to fixed-sized objects composited together, by hardware, with a background.3 Use of the term has since become more general.

One source states:

Sprite

A sprite is the traditional term given to a 2D image displayed in a game

Billboard

... you need to re-orient each particle so that it's facing the viewer. This technique of re-orienting the sprites is called billboarding.

Another source:

Billboarding is a popular technique used in 3D graphics programming. Billboarding allows an object (usually a quad) to always face a given camera. Here are some common uses of billboarding:

– particles – halo surrounding an object – trees rendering

For the particular case of particles, the billboarding is a GPU built-in feature when point-sprites are used (a single point is transformed to a billboarded quad).

Yet another states that both face the camera, but billboards only rotate about their vertical axis (think trees).

Some references specifically for OpenGL:

Live Examples by Three.js/WebGL (though I can't tell the difference):