In APEX 5.1 you can!
Upload an image as a Static Application File. According to the help it should by 64 x 64 pixels, but I noticed it can be larger as well. It should be square though to show up nicely. Don't put it in a (sub)directory, it should be placed in the 'root' folder.
Then edit the Application Definition (either via Shared Components > Application Definition Attributes or by clicking on the "Edit Application Attributes" on the Application Builder home screen. Then scroll down until you see the "Icon File Name" property. Enter the name of the uploaded file there - but don't reference #APP_IMAGES# as you're used to!
Now switch back to the APEX App Builder Home screen ... and your application is there with your custom application icon!
So in 5.1 there's no difference anymore in the presentation of Packaged Applications or your own... they can all look good!