To me, "courier" suggests a niche market. For example, a medical courier might transport samples from a collection site to lab, or it might transport an organ to be transplanted from one hospital to another.

Courier and Delivery(man) are not interchangeable. They have different meanings. Deliveryman usually delivers goods (wholesale or retail). ex. Tomorrow I have to be home because the deliveryman is coming with the new furniture I bought. Courier is used in a more official, legal, business, governmental situation. ex. Courier can be a diplomat who delivers messages. It can also be someone who is a runner of contraband (i.e. drug runner). It can also be a member of the armed services who delivers mail, information, supplies, etc. Someone who delivers documents between company offices., etc.

It's not a case of commonness, it's a case of different meanings. A courier service delivers small packages quickly. A delivery service delivers larger items, and often more slowly. So - you would get a delivery service to deliver a fridge, but a courier service to deliver a memory stick. 'Service' refers to the company. 'Deliveryman' refers to the person that actually does it. BUT - most people won't use a sexist term like that any more. 'Courier' could refer to either the driver or the service.

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