Voldemort never drank unicorn blood through his face on the back of Quirrell's head. Instead, Harry sees Quirrell leaning over the body drinking the blood himself. During the graveyard scene in Goblet of Fire, Voldemort talks about how he still had the power to possess animals, but the animals all died shortly afterwards.
I sometimes inhabited animals — snakes, of course, being my preference — but I was little better off inside them than as pure spirit, for their bodies were ill adapted to perform magic… and my pos- session of them shortened their lives; none of them lasted long….
— Goblet of Fire, chapter 33 (The Death Eaters)
It's likely that by drinking the unicorn blood, Quirrell was preserving his own life.
Keep in mind that contrary to the movie, it wasn't Harry's attacks that killed Quirrell. He died because Voldemort fled his body:
“The servant died when I left his body, and I was left as weak as ever I had been,”
— Goblet of Fire, chapter 33 (The Death Eaters)
Firenze was right, he really was only an inch from death, and it was only a combination of unicorn blood and Voldemort's power that kept him alive.
Alternatively, unicorn blood was one of the ingredients in the potion that allowed Voldemort to take his weakened baby-like form before his body was fully restored. So it's possible that by drinking the blood, Quirrell was somehow allowing Voldemort to create a physical face.