VPSs (Virtual Private Servers) allows hardware to run at a higher utilization rate.
They are cheaper since usually they have less power costs than one physical server.

Their maintenance is still more expensive since it is difficult to patch all virtual servers running different operating systems on different physical servers which still needs separate maintenance. All maintenance on higher amount of systems must be done quick so maintenance time is as low as possible.

It might be difficult to make backup of one arbitrary software. Security issues are more complex in VPS providers than in dedicated server providers. There are two layers of security : both physical server security and VPS security. VPS monitoring might be more difficult than dedicated server monitoring.

VPS requires usually more expensive drivers like multi-port network adapters and more physical connection than dedicated server. There are some more management problems since you will use more Ethernet cables and network equipment like switches.

VPS providers tends to need more IP addresses than dedicated providers. If you have only 100 physical computers with 20 VPS on each it is still 2000 IP addresses. Also, physical system which runs more VPS might have very high number of packets per second (PPS) though network. It is difficult to manage large number of IP addresses, MAC addresses etc.

There are sometimes software licensing advantages when running VPS. Some software licenses are by number of physical processors or physical computers and therefore they are cheaper when you can divide its price by number of VPS on one physical computer.

VPS performances nowadays are very good with overhead less than 5% on some modern virtualisation software like Virtuozzo. But, disk I/O performances might have bottlenecks due to more disk fragmentation.