Specifically, when I add a map service to AGOL maps the labels I created in Arcmap appear. When I add just the layer from the map service to AGOL maps the labels do not appear. Why?
When you add a map service to a web map, the server that hosts that map service draws an image of the service and returns that image to the client to draw as part of the web map. The labels are part of the image, thus you see them in the web map.
When you single out a particular sublayer in your map service, the server no longer draws an image to return to the client. Instead, the server just returns the geometric coordinates of the features in the sublayer. It's now up to the client to draw the geometries based on the returned coordinates. In this scenario, there are no labels.
AGOL does have an ability to draw labels on features, but doesn't have the sophistication of what Desktop can do. Thus, if you need advanced labeling, you'll need to stick with your map service.