In the US, a shipment would involve "sending out" something involving a "common carrier" like a postal or a courrier service. It only applies to goods. I know that in the UK, I've received notifications that items have been "dispatched" and this is understood to mean that the order has been "dispositioned" or "processed through the system and sent out." In this case, they would be similar in meaning. Dispatch, however, has other uses. It would also be used for "sending out" something to a "receiver" that may or may not be necessarily goods-related. For example, a police officer is dispatched to your address if you call for an emergency or to report a crime. A taxi is dispatched to your address if you call for a cab.

A shipment is the product or item being sent; once it is sent it has been despatched. 'Dispatch' is often the department that sends things; where goods are dispatched. Before technology over took them, dispatches in the military were formal written messages. So a brave soldier may have been mentioned in a dispatch.

The difference is obvious. Shipment is goods sent from a consignor to a consignee. Having signed a contract and getting the money you make a shipment or 'they' make a shipment to you when you buy something from an internet shop for example. 'Dispatch' describes a process for sending messages or letters or a message and a letter of its own.