The ArcGIS Server and ArcMap call the same underlying solver. So there should be no difference in how the actual solve happens. The difference is in what settings are on or off between the two different ways to call the underlying solver. Perhaps there is some different restrictions set, or in ArcMap hierarchy is OFF but server has it set to ON, or the points are being located on a different edge or with different curb side properties. So I suggest changing some properties in ArcMAP and seeing if you get the same route as server. If you cannot find any differences, then I suggest you can turn on logging in the server and capture the calls being sent for the route solver and post them here and also post a screen shot of the proprerties of the route layer (analysis settings and/or attribute parameters) from ArcMap and also the route/directions that you get from server.
Jay Sandhu