You have asked a very open question here, but (and I'll let you decide which is a pro or con):
Rules on the server may affect all devices which connect to the email client (pop3 and imap depending etc), such as multiple desktops, mobile phones, tablets etc.
Servers are typically always on so rules occur at 'real' time.
Less likely for local users (desktop users) being able to change the rules (in a business set up).
Server rules can be applied globally (ie, to all user accounts of that domain/server etc)
It may be easier to configure user specific rules on the local PC instead of the server.
You can consider global settings on the server, local / specific settings with the desktop and can have both running rules together (and then consider how you manage this!)
You can change the rules in 'offline' mode with it being created locally.
Rules on the server may not be desired for each local user. For example, moving an email based upon a rule to a certain folder may not be ideal based upon the users job type (I may want the "sales" emails to go to my sales folder as an accountant (as I don't care) but as a sales person, I want it in my inbox so I can't miss it).
My personal opinion is, unless you need access from individual PC level, and are confident to make the changes on the server, then making the changes on the server is probably the better option. One negative though may be, for example, if using IMAP. Let's say you have a rule on your server to move an email to a certain folder based upon your criteria. This then means your web client must sync to all folders!