If someone is alone, there is nobody with them. If someone does something alone, nobody does it with them. They may be happy or sad to be alone. Note that you do not use ‘alone‘ in front of a noun.

1. Again he feels he is alone.
2. I quite like travelling alone, actually.

If someone is lonely, they are sad that there is nobody with them or that they do not have any friends.

3. It will be so dull and lonely here without you.
4. We started it mainly with the aim of helping lonely people.

If you talk about a lone person or thing, there is only one person or thing of that kind present, when you might expect there to be more. Note that lone is always used in front of a noun.

5. In an American suburban street a lone pedestrian is more conspicuous than a lone motorist.
6. Suddenly a lone figure came struggling out of the swamps.