Burners are supposed to use a USB cable with two connectors. With just one, they can't get enough power to burn. Your DVD burner probably came with a cable with two connectors. And at some point someone used it with a cable with one connector, saw that it worked for reading, and put the proper cable in the drawer. They can usually read DVDs just fine with one connector. But every one I've seen needs two to burn.
No you can't hurt a device by using a two-connector cable. All it does is increase the max power the device can use from the USB ports. If the device doesn't need the max, it won't use it.