If the Ashikaga Shogunate treats the Emperor as a figurehead, and now Nobunaga has puppeted both the Emperor and the Shogun, what was the use of his continuous support of the Shogun? How did the benefit to his name of supporting the Shogun differ from that of supporting the Emperor?
Nobunaga indeed had no use for two puppets. In 1568, Ashikaga Yoshiaki used Nobunaga to get back control of Kyōto, and Nobunaga used Yoshiaki to get to Kyōto. Relations turned sour almost as soon as he entered Kyōto, and by 1570, Ashikaga Yoshiaki was part of an anti-Nobunaga alliance openly fighting against him, and by 73, Yoshiaki was expelled from the Kyōto area by force (again).
Nobunaga's relationship with the court is complicated and still debated. There's a question of whether Nobunaga asked Emperor Ōgimachi to retire in 1581 or if the Emperor asked Nobunaga his opinion on the subject (he retired in 1586, 4 years after Nobunaga's death). A couple of decades ago it was even theorized the court played a large part in Nobunaga's death.
However I think the evidence better points to a mutually assisting relationship, at least after the ousting of Yoshiaki. Nobunaga used his considerable resources to restore the courts finances and political functions. The court in term seem to have supported Nobunaga politically. The court, after all, would rule Japan through Nobunaga if he did succeed in his conquest. As described here, the court held firm the power to formally bestow ranks and titles, and so had considerable legal powers. After entering Kyōto, Nobunaga climbed in official rank quickly reaching Third Rank General of the Guard, the highest official military rank (Seii Taishōgun was technically supposed to be special-directive, temporary rank). And except for refusing the rank of deputy-Shogun (which was Yoshiaki's idea) seem to have had good relationship with the court. Nobunaga demonstrated he at least respected the court enough to agree to a peace settlement with Ishiyama Honganji in 1580 when the court ordered it (though everyone wanted peace at that point). He also seemed to have great relationship with Konoe Sakihisa, despite being erstwhile enemies (Sakihisa had sided with Yoshiaki in 1570, also despite being erstwhile enemies), who worked in the court with Nobunaga from 1575 onwards, rising to Daijō-daijin in 1582 (he was Kampaku before Yoshiaki ran him out of Kyōto in 1568). Nobunaga supposedly even promised the Konoe family a province once he finished conquering Japan.