The meanings overlap, but usually:
"Lonely" is an emotional state. It is a kind of sadness or melancholy. It is usually applied to people. "I am lonely" means not only that I am alone, but also that I would like to have some companionship.
"Lone" is a simple statement of fact. It means something is isolated, single, separate, or unaccompanied.
It would be common to speak of a "lone tree" or a "lone rock." In the United States the flag of the State of Texas features a single large star, and therefore Texas is "the Lone Star State." It does not mean that the star on the flag of Texas is sad and wishes it had other stars to talk to, it just means that there is a single star on the flag.
In the fairy tale "East of the Sun and West of the Moon," we read: "So one day, when the White Bear asked what it was that she lacked, she said it was so dull and lonely there, and how she longed to go home to see her father and mother and brothers and sisters, and that was why she was so sad and sorrowful, because she couldn’t get to them."