OK, after finding some help, I've solved the issue. It was an apache config error. I'm still not certain what caused it, but here's how I solved it if it crops up for you.
By using CMD from inside the wamp directory, I saw the error was in the httpd.conf file. I was able to open that file up in my code editor (Notepad doesn't show line breaks, so it's impossible to read), and there I simply removed the offending code.
The steps to solve apache config errors:
Use command line (CMD) to access the wamp apache directory (use x:
to change drives to drive x
if needed) at \bin\apache\apache2.x.x\bin
Run httpd.exe -S
to ask apache to validate the httpd.conf file
Read error that is returned, if any. Identify the line number, file, and error.
Open the apache2.x.x\conf\httpd.conf file in your code editor, go to the offending line, and change the offending code.
In this case, the validation routine returned:
AH00526: Syntax error on line 241 of V:/wamp/bin/apache/apache2.4.9/conf/httpd.conf:
Illegal override option Require
I opened my httpd.conf file, and found on line 240:
<Directory />
AllowOverride all Require all denied
</Directory>
I simple removed the Require all denied
bit, saved the conf file, started Apache, and voila. I don't understand what that option was intended to do, or how it got set, but wamp is running like a charm now that it's removed.