Difference between afp and smb

I had Sophos antivirus turned to scan on access. Could that be the problem?

Could be. Certainly if you turn it off and the problem goes away there's a strong indicator that it is your problem.

Was this installed server-side? or on the clients? If it's on the server it might be overwhelmed with lots of users opening lots of files in succession. It might be better to run your AV on the client side.

When I check the server preferences, file sharing has turned off by itself. When I try to restart it, it just turns off again

That's clearly not normal. What do the logs have to say?

The users get on via afp. Would smb work better? What's the difference between the two?

AFP was designed by Apple and is the native file sharing protocol in Mac OS X.
SMB was designed by Microsoft and is the native file sharing protocol in Windows.

At one time Macs would use AFP, Windows systems would use SMB. That line is now blurred by the fact that Mac OS X can talk both AFP and SMB, and Windows machines can be persuaded to talk AFP.

There are some under the hood differences and in general your Mac clients should use AFP if possible. Whether SMB is more or less reliable in your case depends on why the server is having these issues in the first place - if it's a resource issue (i.e. not enough memory/cpu/etc.) then enabling SMB is likely to make things worse since the server now has an additional process to keep running.