The Apache configuration is the same, and as far as I can tell, all the files are in the right places, and linked properly (php4apache2.dll and php.ini in C:/PHP4/).
When I copied the php4apache.dll file from PHP 4.4.6 into C:/PHP4 (which I'm sure isn't a good solution), it worked.
The checksum for the zip file was correct, and I'm running Apache 2.0.59.