Adding a second wifi router / wireless access point

I have an existing wireless router setup and I want to add better coverage to a certain part of my house. The current wifi covers the area, but the signal is fairly weak and the throughput is commonly falls below what I can sustain through my cable modem.

I have pulled an ethernet cable through the walls, and I am considering just sticking a second access point on the end of it. Connect both ends to the LAN ports of the routers, give the new one a LAN address in the subnet, and it is on the network. Of course I will set the radio to use a different channel than the original access point.

My question is: Can I assign the same SSID to the second device and have the wireless clients auto-magically connect to the one with the best signal or I am better off giving them different SSIDs? Do I have to use Wireless Distribution System (WDS) of some sort even with the ethernet link between them?

Personally I would go with separate SSID's. Every time I have attempted dual SSID's without special hardware it went wonky. Just remember to turn off dhcp on the second router.

It is not necessary to have a different SSID. Having the same SSID and password should provide your users with the need to configure a single wifi access and a seamless change between AP.

What is important though, is that you use different wifi channels for each AP. Ideally the channels shouldn't overlap otherwise you will generate interferences and lower the network throughput. This documentation is very useful to understand how that channel separation should be done.

WDS is not needed as you have a cable between the two AP.

See this similar question and this other even more detailed.