So far, for me, nothing beats Windows own Remote Desktop. It's fast, it is much more visually appealing than LogMeIn and it's free.
Remote Desktop is the fastest, that's true. No one can beat Microsoft when it comes to developing software for Windows.
However, as with other "built-in" Microsoft tools it provides just a bare minimum of what can be needed from a remote control program. Otherwise, we would never hear about third-party tools.
In other words, Remote Desktop to major proprietary remote control utilities is like MS Paint to Adobe Photoshop or Corel Draw.
I think you need a static IP address to use it effectively over the internet (for Dynamic IP the Logmein, etc. programs work fine). You do have to open your router and twiddle the port forwarding -- not difficult.
You no longer need an IP address to connect over the Internet. At least not since 2005 or so. TeamViewer revolutionized this industry and now any remote control software that claims to be good must have such an option. This is how the
remote connection over the internet works.