I disagree with John's assertion that the difference depends on how many colours you use, I believe it's possible to dither using a single colour.

Halftoning is a term used in the print industry to describe how to reproduce varying tones with significantly fewer inks. It's usually synonymous with amplitude modulation halftoning, where a fixed pattern of dots vary in size in order to create the illusion of continuous tone.

Left: two examples of monochrome dithering, firstly pattern dithering and secondly diffusion dithering. Right: two examples of colour halftoning by amplitude modulation, the first with a very large dot size, the second with a smaller dot size that is more convincing.

When colour halftoning, the grid of circles for each ink colour are set at different angles so they don't land on top of each other.

Dithering is a more general term that refers to randomisation or perturbation of colours values or positions or intensity in order to simulate more tones than are available. Essentially both terms try to achieve the same effect, to fool the eye into seeing more colours than are there, and you could argue that halftoning is a form of dithering.

Like all terms they are misused, but to me the difference is that you wouldn't usually talk about halftoning if you were working with images for display on a computer. Whereas, if you were talking about printing then you would usually use halftoning to refer to amplitude modulation (little circles), and dithering otherwise.