Then it's a simple permission problem. I just tried to clear the cache and the files were cleared nicely. Then I tried searching again, and the files were not re-created, which means that the plugin cannot access the cache folder. It was working on your local MAMP server, because the permissions were ok.
If the permissions are set to 777 via ftp on the cache folder then it means that the server administrator has locked the permission changes via SSH. In other words, you might see 777 on the cache folder, but it's actually something different in an SSH terminal.
I think you will need to ask the server admin to unblock the access to the timthumb.php file (unless it's denied in htaccess) or to change the permissions on the cache folder.
You can try different permissions as well, some server hosts do not allow 777. These permissions should also work: 666, 655, 755, 766
755 is a standard for almost every host, you should try that first. You can try that on the timthumb.php as well, it might work.
As for the file name differences: You copied the cached files from your local MAMP server to the live version. That's not going to solve the issue, because the images are re-generated from time to time with a different name in case they are changed. The file names are generated based on the search instance id, image name, timestamp etc.. so that's the reason why the file names are different.Best,
If you like my products, don't forget to rate them on codecanyon :)