"Alone" means unaccompanied, and can be an adjective or an adverb. Sometimes there can be an implication of loneliness (the unpleasant feeling of needing company), but that implication is not necessarily there. "Lonely" means having the unpleasant feeling of needing company. We do not use an adverb for this idea, but you could construct an adverbial phrase such as: "with a feeling of loneliness".

Alone can be an adjective or an adverb. The two words obviously share the same root: lone.

When alone is an adjective, it has the same meaning as lonely.