Here's what I think:

Being lonely means you're missing someone, or something. It means that you feel sad because you lack the company you so desire. It means that you wish someone who wasn't there, was.

Being alone means that there just isn't anyone. There is no one to come and break the solitude. You are the only one, there is no one else there. Just you, by yourself, nobody else, ever.

And by definition, we are alone, in our bodies and heads anyway. So here it is, me alone.

Lonely doesn't sound so bad after all, does it?