A lonely person desires companionship. A lonesome person is lonely in a profound, long-lasting, philosophical, or especially forlorn way. The difference can be subtle where there is one, though, and there is obviously much common ground between the words. Lonely, the older of the two, is safer in serious or formal writing.
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]
[R]arely 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]