For me, I feel that alone is when you simply don't have anyone around you, or if you do, you are not actually associated with anyone around you, and you are wherever it is you are by yourself. I can have 500 people around me, but I can still be there alone. That, by itself, can be positive or negative. If you like being alone, which sometimes I do, then that's fine. If you crave the presence of another person, that's lonely. Lonely can happen anywhere, at any time, whether you're by yourself or in a packed stadium- lonely is more about feeling that there's no one in your life to be with, even if it means you're at home and wish you could go do something with a friend, or wish you had a friend, it's not always about an SO.
I'm having a bout of that, myself, recently, and I need a girlfriend. lol I'm just such a relationship guy that if i'm not in a relationship, i'm unhappy. Currently, I am alone at my computer, but i'm lonely in general, wherever I go.