We use both words in Britain, but they do not mean the same thing.
In British English, 'biscuit' is the more general term. It corresponds more or less to what an American would call 'cookie'. The term 'cookie', meanwhile, is a little bit different. A cookie could be considered to be particular type of biscuit, or it could be seen as a product that's half way between a biscuit and a cake.
In the UK, a biscuit is a small, flat, sweet, fairly dry product. It can be any shape (round, rectangular, oval) and is usually about 5 cm in diameter and 4mm in height. Some are quite plain, whereas others are covered in chocolate or doubled up with a creamy filling. They generally come stacked up in packets of a dozen or more. If you don't open the packet, they'll last for months or years. They're also crunchy - if you bite into a biscuit, it goes 'crunch', with a breaking sound.
For BrE speakers, a cookie is larger and softer. Cookies are always round, and approximately 8 cm - 12cm in diameter, sometimes larger. They have a much looser texture, often including oats, and tend to be studded with pieces of fruit, nuts or chocolate pieces. But the crucial thing about cookies is that they aren't crunchy - they're moist, and there's silence when you bite into them. They're often sold singly, or in a set of no more than six.
The exception to this is a biscuit called a 'chocolate chip cookie', which looks a bit like the second category, but actually belongs the first.