It essentially removes having to pass data through an extra step (HTML) and lets Flash directly access data you have in a database. So rather than depending on the HTML page to update your flash data (you'll have to refresh the page, or work some tricky Javascript), you can just have Flash go out and query the database again directly.
Another bonus for me is that when my ColdFusion <cfquery> call returns a Query object, Flash understands that and can can process that data easily (and without giving me headaches

).
Here's a template that I use whenever I set up my Remoting .fla's:
Code:
// Set up the gateway to the Flash Remoting component
NetServices.setDefaultGatewayUrl("http://[site domain]/flashservices/gateway");
var gatewayConnection = NetServices.createGatewayConnection();
// direct the server to the ColdFusion component file it's using
var server = gatewayConnection.getService("[ColdFusion component path]", this);
// the server calls a function inside the component file
server.[ColdFusion Function name here]( [arguments] );
[ColdFusion Function name here]_Result = function(rs) {
// get the number of results.
max = rs.getLength();
trace('returned ' + max + 'results');
}
I'm not sure it's the fastest way to transmit data from the database to the flash app, but it works fast enough to satisfy my clients...