I recently ran into an issue where a user would insert an SD card, CD, or other removable media and then the AutoPlay (note this is different from, though related to, AutoRun) dialog window pops up briefly for a split second and then closes itself automatically. At first I though the ESC key or something was stuck down, but that wasn't the case this time. In addition, it would even automatically exit the AutoPlay control panel window as soon as that opens.
It turns out that there are a number of software programs that could conflict with Windows AutoPlay. Examples include some virtualizing software and even some CD burning/CD writing software. In our case, it turned out to be an issue with a Motorola phone driver, of all things, that was loaded in the system (Motorola MotoHelper Driver). If that fits what is happening to you, you would want to uninstall the driver from the Programs and Features control panel.