Add Web Reference is the old-style, deprecated ASP.NET webservices (ASMX) technology (using only the XmlSerializer for your stuff) - if you do this, you get an ASMX client for an ASMX web service. You can do this in just about any project (Web App, Web Site, Console App, Winforms - you name it).
Add Service Reference is the new way of doing it, adding a WCF service reference, which gives you a much more advanced, much more flexible service model than just plain old ASMX stuff.
Since you're not ready to move to WCF, you can also still add the old-style web reference, if you really must: when you do a "Add Service Reference", on the dialog that comes up, click on the [Advanced] button in the button left corner:
and on the next dialog that comes up, pick the [Add Web Reference] button at the bottom.