I'm fairly strongly introverted, and in my mind, the difference is this:
Being alone: Quite simply, you're the only person around at the time. This may or may not be intentional or desired. For instance, I'm sitting in my room alone right now on the computer, about to play some video games, and I have no complaints. Being alone isn't necessarily a bad thing, or a good thing.
Being lonely: This is where you wish you had somebody around to connect with, and don't. Sometimes, it is possible to be lonely even though there are lots of people around. You're at a party, and nobody's talking to you, or whatever. Being lonely is, by definition, a bad thing, because it is a state of being unfulfilled emotionally.
I remember once when my family was on vacation, I was on my own at home for almost a week. At one point, I realized that I had not talked to another human being in three days. I wasn't lonely, but, other than my cat, I was completely alone. I wasn't unhappy; I was actually having a great time on my own.
I can also remember times where I have been lonely. I'm a 33 year old single guy, and I've never had a significant relationship with a woman, for instance. Sometimes, this makes me feel quite lonely, and this can happen even if other people are around; it has nothing to do with the number of people in the room.
I have no idea if anything I said made sense or helped or anything, but that's my two cents.