I DON'T BELIEVE IT!!! I was stuck for two days and NOW it struck me!
I had the same MAC for client and server! The router did not mind, DHCP worked, requests were coming and going between LAN and internet. BUT when the two devices tried to connect to each other - obviously - the connection failed, badly!
SurferTim, thanks for the super fast reply!
If anyone wants to use the client code, I have made a sketch that depending on serial input connects to a different server / command combination.
Sorry, false alarm!
Edit:
Please be advised that this phenomenon is NOT AS TRIVIAL as it may seem! MOST of our codes are running on the same "random" 0xDE, 0xAD, 0xBE, 0xEF, 0xFE, 0xED MAC address, my year-old server, the days old client, bildr code, SurferTim server AND client code, Arduino sample code (IDE v1.0.1) they all have it.
Most samples of server/client code will probably NOT work for direct Arduino-Arduino communication unless you CHANGE THE MAC ADDRESS!