There's no special reason why one uses RTMP vs. AMF. Data
management service can also use AMF and Remoting can also use RTMP.
The only real requiement is that Data management service needs to
be able to receive updates from the FDS/LCDS server. This can be
achieved by using a channel that is capable of receiving pushed
updates from the server (eg. RTMP) or a channel that is capable of
polling for updates on the server (i.e. AMF-polling). Therefore,
when you use AMF with Data management service, you need to have
polling enabled with a polling interval.
In terms of why AMF works but RTMP doesn't is that AMF
channel uses AMF format over the HTTP protocol whereas RTMP channel
uses simple TCP socket (and not HTTP). Therefore in some
environments, RTMP will be blocked by the firewalls but AMF won't
since it's like regular HTTP traffic. In this case, you can use AMF
polling instead or use RTMPT which is new in LCDS 2.5.