Connecting Airport to switch to extend a network
If you have them set up as a wireless extender and then connect them with a wired connection, you are creating a network loop.    



The drawing is a bit rudimentary, but it makes the point.  Whether you are connected over a wire or a radio, you have a connection between one Airport and another. When you make the second (wired) connection, you create a second path back to the Airport.  

So, if you imagine a client device connected to any one of the two lower switches in the picture, which path should the switch send data destined for the client?

That's why things go awry.

"Enterprise" grade network switches have something called STP (Spanning Tree Protocol) also known as 802.1d which prevents this from happening.  Looking at the Tech Specs page for the Airport Extreme, it appears that 802.1d is not supported; not surprising since it's rarely on consumer grade devices.

To fix it, you need to leave the wired connection in and turn off the "Extender" mode.  Set it to be a Wireless Access Point instead.