The server is what handles the world, and mechanics, of UO.
The client is what allows you to access that world.
When you do something in the client, the data is sent to the server. The server then performs the bulk of the necessary calculations, and sends data back to the client. That data is then represented by the client.
A client patch, in effect, modifies the interpretation of the data sent by the server, in an intended manner. For example new graphics, or localisation strings, or adjustments to the interface of the client.
A server patch, or publish, modifies the "world" and its mechanics. A turn-in NPC, and its associated mechanics, is a server-side object (imagine what would happen if that was all worked out client-side
).