The map service layer gets drawn differently depending on how you add it to the map.
If you add it without an index number for a layer (e.g., .../0) it means you are adding the map service and all sublayers inside the map service. In this case, when the map service layer is drawn, the map viewer of arcgis.com requests an image from the server of the layer. The server creates the image and returns it to the map viewer. The server is doing all of the work to draw the layer.
When you add a layer to a map with an index number (with the /0), you are now bypassing the the image drawing on the server. Instead, when the map viewer requests that individual layer from the map service, the server returns the actual coordinates of the features in the layer. The features are then drawn by the web browser (or other client) as graphic elements on a "graphics layer" of the browser. This graphics layer draws on top of everything and thus draws over any other map service layers drawn as images.
This post may help as well Who can make this  ESRI'S reply simple and easy to understand ?
While it describes the behavior of shapefiles, the same applies to map services drawn with an index number (effectively, you are drawing a map service as features).
Hope this helps.
Mike
