I figured this was a common issue, so I searched VMware's KB and found this article: Unable to upgrade existing VMware Tools (1001354). I ran through the section on Windows 8, but none of those steps applied to my installation. The last section (finding and deleting all occurences of vmware in the registry) was a little more than I needed. I didn't want to corrupt all of the VMware software I've got loaded (PowerCLI is a MUST!). So I tried something different.
In the error screenshot, you'll notice that the source field references a CLSID (it's the data that begins with {8AE43F1). The CLSID uniquely identifies the application that you're working with. I decided to search the registry for any data that matched the first portion of this CLSID. Here's the result of that search:
You can see that the search turned up an entry that referred to VMware Tools and the path to the installer. I decided to delete this whole key and then restart the VM. (This is where timid bloggers will say "make a backup." I say just delete it. I'm bold like that.)
Success! I was able to install VMTools without error after getting rid of that key.
Now VMTools is current and running. Don't waste time wondering if this is a Windows problem or a VMTools problem. You'll drive yourself mad in the process. Just fix it and move on. Nothing to see here.