Broadly, I’d say in the UK we use the word “biscuit” in the sense that the word “cookie” is used in the US. For us Brits it means, generally, a small, flat (ish), sweet baked good that’s hard and rigid. Something you’d nibble and dunk in a cup of tea. They can be round or square or rectangular. They aren’t made with yeast, but could be made with chemical raising agents such as baking powder or bicarbonate of soda.
We do also use the word “cookie” in the UK, but I tend to use them fairly synonymously.
Which probably isn’t very helpful, as, for reasons revealed by a spot of etymology, they are technically somewhat different.
The word biscuit is from the 14th century Old French word bescuit. It literally means “twice-cooked”, or twice-baked and derives from bes bis + cuire to cook, from Latin coquere. The Italian word biscotti has an identical meaning. And indeed, the classic biscotti, the hard slices that you dunk in a digestivo or coffee – biscotti di prato, cantucci, cantuccini and tozzetti – are the best example of this process, as you bake the dough in a loaf form. You bake the dough in a loaf form first, then slice it, then bake the slices to crisp them up.
The word cookie on the other hand derives from the Dutch koek, meaning cake, in a diminutive form: “little cake”. Cookies are more technically, therefore, derived from a more cake-like dough or batter, often dropped in spoonfuls on a baking sheet, not manually moulded like a firmer biscuit dough.
Anyway, I’m going to keep my biscuits and cookies in the same category. But I will put crackers – basically savoury biscuits – in a separate category.