Almost all, if not all, of the trains between Moscow and St. Petersburg are either overnight trains with sleeper compartments, or fast Sapsan trains. These trains tend to run quite full, and I don't know if there are any unreserved trains on that route anymore. So, even if your ticket were technically good on these trains (and I'm not sure it would be), you may not be able to find a train that has space. If you intend to go from Amsterdam to St. Petersburg and then go from St. Petersburg to Moscow a few days later, I'd make sure to be ticketed that way, and get your reservation for either the Sapsan seat or the sleeper compartment of the St. Pete to Moscow section in advance.
More on Russian trains here, from The Man In Seat 61: http://www.seat61.com/Russia.htm#.U8Q5e7GGe8A