You have to pay for vSphere with its various modules and extra features but not to use the vSphere Client to connect to a free ESXi.
I think where you may be getting the license message from is although ESXi is free, you still need to request a free license key from VMWare.
Login to your ESXi box with vSphere Client and go to Configuration -> Licensed Features -> Edit.
If you are set to evaluation mode, that is what you are getting the license warning from.
VMWare should have emailed you a license key when you signed up on their website to download ESXi. If not, you can go through the download steps again and the license key should be on one of the pages.
For me, if I go to https://www.vmware.com/products/esxi/ hit Download, login with my free VMWare account, then on the page with all of the download links, at the top of the list is my ESXi License.
The reason you are seeing the license message about vSphere is that in the Evaluation mode, some of the extra features that are only available with vSphere are enabled, once you enter a free ESXi license, those will be disabled and you won't get prompted anymore.
Also, you can use the vCenter Converter in the standalone mode (runs off of your workstation) for free with ESXi. This tool is immensely useful for moving VMs on and off of ESXi. http://www.vmware.com/products/converter/.
"When I made the clone I'm pretty sure I copied the .vmdk and .vmx files to a new folder via vSphere Client datastore, and then hit 'Add to Inventory'"
This isn't cloning but just copying, am I wrong? It sounds like you went to the datastore, right clicked on the VM files and selected copy? From you description you have 2 servers with the same SID/CMID. Also if you only copied those 2 files I think you are missing a few there. http://www.vmware.com/support/ws5/doc/ws_learning_files_in_a_vm.html
I would have right clicked on the server in vCenter and done actual cloning, not copying, from there. You can customize the clone after that to create a unique SID/CMID (sysprep).
Best Answer
Sounds like a diskless or stateless deployment, this may shed some light on what you're experiencing. Ideally your scripts should reside in a datastore, not on the root.