I installed a product that said to set the "Log On" user (in the Log On tab in the Service Properties option) as the network user. The service would not start. I then changed the "Log On" user to Local System Account, and the service started. In general, what is the difference, and what conclusions may i make about the two users?
The Local System account is the computer's operating system's own security context. It has all the rights of "Administrator" users and more.
More than likely the account you're naming that is not working properly doesn't have permissions to access some resource needed by the service program. You could put that account into the local "Administrators" group as a "quick fix" and see if the service starts. If it does, then the Right Thing(tm) for you to do would be to track down exactly what the permission problem is and recify it. (Sadly, too often the service user just gets left as a member of the local "Administrators" group and nobody ever tracks down the source of the problem...)
In terms of tracking down the source of the permission problem, if it does turn out that the service starts properly with the user as a member of the "Administrators" group, remove the user from the "Administrators" group and monitor the service startup with the Process Monitor tool and you'll likely find an "ACCESS_DENIED" entry that leads you in the right direction.
Does the other user have "log on as a service" enabled? If not, that's probably the issue.