Explain Host- and Domain Name in DD-WRT

Could someone please explain in detail what Host Name and Domain Name are used for under Internet Setup in the DD-WRT router?

I've had them blank and it works perfectly, so did it for a friend of mine. But I'm curious to know what they are used for and when.

Here is a screenshot of the interface containing these settings as well as the available in-built documentation (which is very sparse about these two):

The more details, the better the answer!

Some ISP's probably expect you to set set up everything on a computer that is directly connected to the network through their modem etc. And perhaps because of security, or laziness some of them would then require your computer's name to match a specific one that they assigned to you. So this way if you connect to them through a router instead, this would allow you to set your router with the settings that they are expecting to see coming from you.

This is the "DHCP host name", which is an identifier sent to the DHCP server. This, combined with the MAC address of the interface making the request would positively identify the DHCP client.