The problem with LogMeIn is that it constantly is polling the main server, which adds a lot of Internet overhead. Its not really noticeable to people with high speed connections, but some of them with slower DSL connections really get affected. Plus, the added resource usage is really too high, IMO. Having a program sitting idle in the tray using up ~60MB of memory sucks.
The only thing LogMeIn has going for it is the easy of configuration. However, setting up Remote Desktop coupled with No-IP really isn't that hard. Most of us know how to forward ports, so forwarding the one port is nothing, and setting up a free No-IP account and installing the client is also childs play. And once you know how to do it, it is extremely easy.
Plus, with Remote Desktop you get Shared Printers, File Transfers, Shared Drives, Copy and Paste*, Sound. All free things with Remote Desktop that LogMeIn doesn't provide unless you pay for the Pro version.
I'll take the slightly more complicated setup for the savings in resources and more functionality anyday.
*Yes, I know you have the very basic transfer clipboard option with the Free version of LogMeIn, but it is a pain in the ass to use. The ability to just Ctrl-C and Ctrl-V in and out of the host/local machine is extremely nice.