LogMeIn has a basic free service. GoToMyPC only has paid services. Most people use the services simply to remotely access a computer, such as accessing your work computer from home, as if you're sitting at the other computer. LogMeIn Free does this just fine, and just as securely (non-hackable, etc) as the paid versions. The paid versions (GoToMyPC or LogMeIn Pro) add things like remote printing (print while you're connected to your office computer, and the printout goes to your printer at home), sound (hear the sound when you get a new email or watch a training video), and file transfer. Sure, these are nice, but they may not be needed in your situation. (LogMeIn Pro also adds "Mini Meeting", a feature similar to one GoToMyPC charges extra for, named GoToMeeting, though limited to one user at a time.)
A big factor, however, is that LogMeIn handles computer accounts similarly, whether paid or free, and a single user account can handle multiple computers. For example, you can have a Pro account to access your office computer. Your employer may see the need and pay for it (it's only $12.95/month or $69.95/year). But you can then install the free version on your home desk computer (check your home email), your media center computer (schedule a TV recording, download a video you just heard about before you forget about it), and your parents' computer (to provide support). All of them can be handles through a single account - you just log into the website and pick which computer you want to use. If you need to access a lot of computers (server support from home without complex VPN configuration), you can create groups to sort them (home, work, servers, family).
More useful in a business environment than at home, you have a lot of options, say, creating user accounts (for telecommuting office workers) while retaining administrative privileges. And there is a support version so users can request support and you can help them, without even having the software installed in advance. Administrative and reporting tools are also available.
But the real key is simplicity. My sister has a home business in Georgia. My mother handles her accounting, and has the accounting software installed on her computer at her home in Tennessee. If my sister needs to look at something, she can use LogMeIn Free to check it, and they can save big bucks because they don't need an expensive multi-user accounting package.
My mother called me with a question on laying out a flyer. I used LogMeIn Free to access her computer and see what she was trying to do; I could then walk her through what she wanted.
But it is safe. It requires several sets of usernames and secure passwords. To access a computer, I first have to log into the LogMeIn website with an account that is authorized to access the computer in question. (I can use my own username and password since my sister's account has me as an administrator.) I then have to log into the computer with a local username and password - I can't log in remotely unless I could log in physically. If the computer accounts don't have passwords, a substitute passcode will be required. Though it does not have the restriction of Remote Desktop - the computer login does not automatically log me into the computer; I can log in or out as necessary with any credentials I may have, all under the same remote session. And all information is sent encrypted.