Alibaba is more wholesale-focused, and products listed there are generally sold in bulk. AliExpress is more geared towards the average customer, where items can be purchased one at a time, much like eBay or Amazon. Alibaba themselves describes the difference this way:
Alibaba.com is the leading platform for global wholesale trade serving millions of buyers and suppliers around the world. Most of the members on Alibaba.com are manufacturers, trading companies or resellers who trade in large order quantities.
AliExpress is a global retail marketplace offering quality products at factory prices in small quantity. Your order amount can be as low as 1 item and you can pay safely using Buyer Protection service.