A lonely person desires companionship. A lonesome person is pitifully forlorn or dejectedly lonely. The difference is subtle, though, and these adjectives are mostly interchangeable. Lonely, the older of the two, is safer in serious or formal writing.
Lonesome is chiefly an American and Canadian colloquialism, but it probably has Scotch-Irish origins. It often connotes an all-encompassing, philosophical loneliness. Sometimes, though, it just means desiring companionship. It also often describes sounds (especially music) or landscapes that are analogous to or reflect a lonesome state of mind.
Examples
Legend by then had fructified Chapman … into a mythic, apple-spreading American nomad of the lonesome frontier. [Wall Street Journal]
In the rolling countryside along the Minnesota border, the lonesome, dusty roads seemingly outnumber the people. [Vancouver Sun]
Two lonesome, studly cowboys ride the range at the foot of the snowcapped Rocky Mountains … [Salon]
… rarely has the lonesome plains of the Canadian Prairies looked so bleakly terrifying. [Toronto Star]
So once again Dayne trudged forlornly to the sidelines, prepared to stand off by his lonesome, where he could contemplate his naval in solitude. [New York Daily News]
