Re: Free for non-commercial use?
RDP is an option as you described but "far more secure"?? What are you smoking?? Teamviewer's encryption is as good as your password, and you can make that as complex as you like, while RDP requires, as you pointed out, opening and forwarding ports on your router (not great for security).
Also if you want to have access to more than a handful of machines inside a router (like a small business, for example) you have to (a) use fixed IP addresses (b) set up different ports for each PC, which IIRC involves registry editing on the client side and some really tedious messing about on your own; AND (c) most user-grade routers only allow for a limited number of port forward rules. Sooo... Teamviewer wins hands down. Oh and also (d) RDP is not implemented on Windows Home, and (e) when you connect to the PC the user's screen goes blank so you can't use it as a remote coaching or teaching tool, only to do things *for* the user.