First, well-known port numbers are "well-known". They eliminate some guess-work as to what port a given service is listening on.
Second, ports 1024 and below can only be opened by
root. This gives an additional level of "trust" to these services. I.E., the service I'm connecting to on port 22 must be running (or have been started) as root. If a service is running on 1234, it could be any user that has access to the box that opened and started a "program-that-acts-like-SSHD-but-intercepts-passwords" program. (This assumes that there is no NAT or other redirections in place - see mmi's answer.)
Also agreed with a point made by mmi - I wouldn't unnecessarily run a program as root just to obtain the desired port number. If a program running as root were to be compromised, the entire system is then compromised. However, if a program is already running as root for other needs, may as well use the proper port number.
Some people will advise against using the well-known ports in an effort to "hide" common services that might be used for exploit attempts, but I consider this to be only "security by obscurity" - and advise against doing such.