As he prepares to turn over the reins of the Service Employees International Union, Andrew L. Stern is at the apex of his political influence. He helped lead the push for the healthcare overhaul that became law last month. He has visited the White House 38 times in President Obama's tenure, far more than other labor leaders. Obama named him to a new deficit commission. But underlying Stern's glittering profile—a well-connected visionary seeking to revive organized labor and lead the liberal vanguard—is the complicated reality surrounding his departure, which he is expected to make official at a meeting Wednesday.