Here's some thoughts on the benefits of introducing a server-client system. Of course, there would have to be a cost analysis too because servers ain't cheap. A data migration policy would also have to be devised, although modern server software does facilitate this.
Software upgrades take place on the server, not on individual machines. You don't have to individually upgrade your firewall or AV every few days. Nor do you have to download patches individually. Maintenance work can be scheduled for silent hours when the network is not being used, often automatically and without human input. Less time is spent synchronising individual computers. Maintenance operations can be automated with script-based and policy-based management tools. The server's "health" (and that of the network) can be monitored easily. Network bottlenecks and other limitations are easier to identify because software designed for this purpose is available.
Data distribution can place enormous demand on one computer if that machine stores the bulk of the data that is required. Instead of allowing the computer user to operate the machine, it spends a disproportionate amount of its resources accessing and supplying data to other users. A server distributes the data exchange loads evenly because you don't just have one terminal doing all the work. It is possible to install a server with a higher capacity cable to a router or switch, thus ensuring that individual terminals are not overworked. Servers are designed for one purpose and operate software which is also dedicated to this, whereas stand-alone workstations have software such as XP which is designed for single desktop use. This also facilitates the speedy transfer of data and enhances productivity by turning "wait" states into action. Data is always available; you don't have to hunt around for the right computer. Data transmission rates vary according to system usage, but it is commonly accepted that performance deteriorates once a peer-to-peer network exceeds between 5 and 10 work stations.
Backup is more certain and easier, because you just need to backup the one database, not several. It is also easier to restore the system in the event of a disaster, because there is only one batch of data to restore. If a client computer fails, the main server can easily restore lost data.
Security is easier to monitor because server software is based on "Kerberos", a much stronger system than that of work stations. Physical security is easier to maintain because only one machine needs to have enhanced protection. Floppy and CD drives on client computers can be disabled, improving security. It is also easier to make files available on a need-to-know basis.
A server can host web pages for employees or customers and it can co-ordinate staff activities more easily, such as holidays. Everyone can readily see what's been arranged. An internal mail system and even internal video conferencing become possibilities. Remote connection is also facilitated.
There are also scalability benefits in that additional servers can be added as the company grows with no impact on the rest of the system.
With peer-to-peer networks, you need to place a lot of trust in your network users' technical competence. The introduction of a server reduces the training commitment for end-users.