I'm using plugin version 1.1, which worked fine since December. Here's syntax my former developer created (which was verified by Hostgator's support staff and by me):
<img class="h-pic" src="http://www.punditmaximus.com/wp-content/uploads/2012/11/download-img.png" alt="" />[/filedownload]
[filedownload file="http://www.punditmaximus.com/wp-content/uploads/2012/12/Neubauer_On_NJ_Master_Plan.pdf"] <img src="http://www.punditmaximus.com/wp-content/uploads/2012/12/down_img.png" alt="" /> [/filedownload]
To help with troubleshooting, Hostgator's staff checked everything they could think of (links, files are in correct locations, he could initiate downloads without plugin, there were no changes in web hosting service since December,...), but he was unfamiliar with WordPress plugins. He was skeptical that Apache server would cause any problem because they have always used Apache, and the plugin worked fine previously. Hostgator ran out of ideas, but he couldn't read the plugin's code.
The various plugins I use on the website are all the same ones and same versions I used since December, when they worked fine. I would be glad to give more specifics if that will help isolate the cause.