Cities naturally 'grow' tiles with culture points. The map tiles around your city become the colour of your civilisation, which means that your city can 'work' those tiles. A city also has the ability to purchase these tiles at a cost of gold.

You can join cities by linking their workable tiles together. I understand that doing so gives a territorial border, but are there any other reasons for doing so?

I want to build cities along the coast, which will be joined by harbours, but I am unsure if not linking their tiles will cause me problems.

The image below illustrates three cities that are joined by workable city tiles: