I've used both LogMeIn, and TeamViewer, and in my opinion, they serve very different purposes. I like LogMeIn for my parents PCs because I can just say I'll connect and give them no instructions on how they let me connect. Its all saved in my account, and it just works.

TeamViewer is a fantastic, lightweight remote desktop client, that I use primarily when I know its a one off issue I'm going to be helping someone with. What I like is that you can use it w/o doing a true install, it runs locally still, but there is a lightweight client that is saved apparently to temp files, and is deleted after you close the session. TeamViewer is about as simple as you can get - it will ask for a 9 digit numeric ID and a 4 digit pin that changes every time you run it. That's all you need to connect.