Pillsbury Co. Monday named Philip L. Smith, head of rival General Foods Corp., as its new chairman, president and chief executive effective next week. He will succeed William H. Spoor, who returned from retirement last February on an acting basis. Pillsbury, the nation's fourth largest food conglomerate, has been looking for a new top executive since then. Smith, 54, has been chairman of General Foods, the largest food company, and vice chairman of Philip Morris Cos. Inc., which acquired General Foods in 1985. While some speculated that Philip Morris might have been unhappy with Smith's performance at General Foods, others suggested Smith merely wanted to run his own show. Analysts have said Pillsbury - maker of Green Giant vegetables, Haagen-Dazs premium ice creams and Van de Kamp's frozen fish and parent of the Burger King fast-food chain - has been lacking clear and vital leadership over the past few years.