Hardware: Rack mounting, enhanced heating, cooling, and electric supply. None of these are required to have a server engineer, but to run several of them efficiently usually leads you to run some kind of dedicated server infrastructure.
Storage: Server storage usually runs on RAID, which takes a little knowledge to manage and maintain.
Backups, Recovery: If your server is accessed by multiple people 24×7, then you’ll need specific equipment, software, and processes to back that data up manually or automatically while minimizing or eliminating downtime for your users.
Security: Data on servers is accessed by multiple groups of people, each of whom has different access rights and privileges. Defining those groups and setting up appropriate data access is task-specific to server engineering.
Server applications: Web, Database, Email, etc. Supporting these applications requires specific processes that you won’t usually encounter in desktop management.
Again, if it’s only one server used for data storage, then it’s quite likely that your desktop management group can handle it (if they have time, dedication and inclination). In general, if you end up with multiple servers running dedicated applications, then you’ll have enough work to justify a dedicated server engineer.