you are right about both things.
In ESXi the "Download vSphere Client" link on the webpage of an ESXi server links to the VMware site: http://vsphereclient.vmware.com/vsphereclient/2/5/8/9/0/2/VMware-viclient-all-4.1.0-258902.exe.
Old versions of VMware vSphere or VMware Infrastructure require their own versions of the client. So you should keep them installed for all versions of ESXi servers you still have running.
I find this very inconvinient of not able to use the new vSphere Client 4.1 to connect to VMWare ESXi 4.0, atleast it should have been backward compatible. Don't know the reason why VMWare didn't work on these lines.
I don't know why the new version of the client is not compatible with previous versions of the server software. If you start the vSphere client v4.1 and try to access a ESXi 4.0 server automatically the version 4.0 client is started. If you don't have it installed it will be downloaded and installed first. So if you have them both installed you don't have to think about starting the right version of the client.
Please close this thread though so far I have not got the correct reason to my question. Will just be waiting for future releases or else continue with both versions of vSphere client