jimmie,
Can you give me/us more information. Of course, in my opinion, RDC is always better than LMI (based only on the overall package). LogMeIn is free, RDC is free. But, LMI ignition is not free and to compete with RDC, you would need it.
If you are at home and remoting into your network, there isn't a whole lot of difference. You bring up a browser, go to LogMeIn, find the computer on it that you had earlier installed the host software and in. I suppose the 45 seconds that takes is not that bad. With LogMeIn ignition, it's very fast. Both methods with LMI are very secure, although you do have the middleman/woman. If you have several computers, you must download the software for each, certainly not a big deal once you have done it.
With RDC, the experience is much richer, IMO. RDC version 6.1 comes with WIN 7. You can download v6.1 for XP SP2 from Microsoft as it has been back ported. Remote Desktop Gateway or RD Gateway is a role service which started with servers 2008. This allows a HTTPS connection allowing a SSL encrypted connection between the remote computer and all of the computers on the network. The RDC client is then configured with the RD Gateway and other information so that you can click on it and connect in about four to five seconds. With the RDC client, each time you add a full qualified domain name FQDN to connect, it is remembered in the dropdown list so that you can easily connect to that computer again with your login name and password remembered (if you wish).
Another huge benefit over LMI when on a network is you can instantly connect to any other computer on the network and, in fact, log from that computer to another computer.
Again, though, can you refresh my mind on how you are using the two technologies?