@usernamenotation, you can let everyone know your post was directed to a specific person so that they have a window into the conversation.
If you are familiar with Facebook, using Twitter is a bit like having a website made of just your profile status and your wall. These are my two favorite features of Facebook, so perhaps this is why I enjoy Twitter. The main difference is that you never post on someone else's wall, your posts show up on their wall if they choose to "follow" you (subscribe to your updates). I've never really used my phone to twitter (yes twitter is also a verb) or receive updates, but supporting posts from multiple platforms is a big selling point for some users.
My Twitter page is here and I also include a feed viewer gadget here on my blog to display my last few Twitter posts. If you have an account post it below!