The destroy()
method cleans up resources so they can be released. When the whole JVM will be shutting down, it's not as critical to release all resources before shutting down, but it's always a good idea to do The Right Thing even when it is not strictly necessary.
Depending on the threading model, if you leave yourself as a mouse listener, then you will still be notified if mouse events occur. If there are multiple Applets in the same JVM and only one Applet is ending, then you can leave threads in a funny state if you leave a listener to which no action will be taken. It's possible that you can lock up the other Applets by doing so.
EDIT:
By threads in a funny state, I mean (for example) if a listener whose Applet thread has stopped is queuing messages to a queue that no-one is reading from, then eventually the queue will fill up and the dispatch thread will block. (In more detail, let's assume the listener does nothing but queue the message and that there is a thread in the Applet -- now stopped -- that reads from this queue. Once the queue fills, it will block!)
With a mouse listener, specifically, you may well be safe, assuming when the Applet quits it is no longer visible and can no longer receive mouse events. With a different sort of listener, however, you could get into trouble. Always doing the right thing, even when it is not truly necessary, gets you in the habit so you don't forget to do the right thing when it's actually critical. :)