Alone:
without anyone or anything else, separated, isolated.

Lonely:
feeling an emotional response to lack of companionship.

They tell me I look lonely.
I tell them I'm not.
They point out the fact that I'm sitting in a corner with a 5-yard radius of open floor.
"Yes," I say, "Isn't it lovely?"
They tell me I can't be happy when I'm alone.
I ask if they'd like to place a wager on that statement.
Confused, they walk away.
Now I'm alone.
Now I'm happy.