What is the difference between "mince" and "dice"?

Mincing produces smaller, more irregularly-shaped items than dicing.

Dicing is generally uniform, usually 1/8 to 1/4 inch on all sides, kind of like a tiny cube.

The best way to explain the size difference is visually, check out this link for a great picture near the top.

From Cuisine at home:

For chopped food, think of gambling dice, roughly 5/8-inch cubes. It’s a good cut to use when making dishes that cook awhile, like stews, soups, and stocks.

When dicing, keep the size of a pencil eraser in mind. You want cubes 1/4- to 3/8-inch on a side. If an ingredient is to be sautéed for short periods of time or eaten raw, as in salsa, then dice it.

Mincing is just cutting food into tiny bits. When you mince garlic or shallots, the small pieces spread throughout a dish, permeating it with flavor. To mince, first roughly chop or dice the food, then rock your knife back and forth over it until it’s small.

It may be an issue of locale - in British English recipes, minced would be synonymous with ground from a American English recipe. Diced stays the same (British recipes may clarify with "finely diced" or "coarse diced")