Under what circumstances would a user be better of using a time sharing system rather than a PC or single user workstation?

The benefits to this are the same as you are using right now reading this page. A time-sharing system (what we call a server nowadays) provides many many people the opportunity to use the system. Technically, even a standard PC is a time-sharing system, because you can run many different programs on it at a time (think Browser, iTunes, Anti-Virus, etc.).

The old mainframe systems had a card reader and an output. Only one set of cards could be run through at a time. So, if payroll was running through all 14000 employee's paycheck calculations, you had to wait until they were done. Modern mainframe systems have some time-sharing capabilities by allowing multiple processes to run on the system, but those processes are virtualized, rather than a true sharing system. However, the details of that can be left to others.

There are benefits and downsides to every type of system. In a mainframe-style system, all resources are available to the process being run, which benefits large data crunching operations (like payroll). The downside to a mainframe-style system is that only one process can run at a time in each virtual machine.

In a server-style system, all resources are shared among all processes. If a process "runs away" by using more processes than it should, then the other processes suffer, which the end-user sees as a server responding very slowly and/or "timing out". The upside to a server-style system is that you can serve web pages to thousands of people, while at the same time handling thousands of pieces of email.