It seems to me that having the Dark Lord inhabiting the back of your head would have some practical effect (other than requiring a turban). Was Quirrell impeded in any way? Was his ability to do (some or all) magic improved or altered by the Dark Lord possessing him?
Quirrell and the Dark Lord remained two separate minds.
The only way Quirrell and the Dark Lord could communicate was verbally, since they spoke their words to each other aloud even when it would have been an advantage to communicate telepathically if they were capable of doing so. In addition, it's clear that they don't share intelligence,
“He lies … He lies …’
‘Potter, come back here!’ Quirrell shouted. ‘Tell me the truth! What did you just see?’
The high voice spoke again.
‘Let me speak to him … face to face …’
‘Master, you are not strong enough!’
‘I have strength enough … for this …” - Harry Potter and the Philosopher's Stone, Chapter 17 (The Man with Two Faces)
The Dark Lord is likely unable to move Quirrell's body by himself.
When they're trying to get the Stone from Harry, the Dark Lord keeps insistently commanding Quirrell to seize Harry and to kill him. If the Dark Lord could control Quirrell's body, he would most likely have forced Quirrell to seize Harry. It's unlikely he'd be asking Quirrell to comply with such an important step toward his achievement of immortality and defeat of Harry Potter if he could simply control Quirrell's body physically and make him do it.
“Seize him! SEIZE HIM!’ shrieked Voldemort again and Quirrell lunged, knocking Harry clean off his feet, landing on top of him, both hands around Harry’s neck – Harry’s scar was almost blinding him with pain, yet he could see Quirrell howling in agony.
‘Master, I cannot hold him – my hands – my hands!’ And Quirrell, though pinning Harry to the ground with his knees, let go of his neck and stared, bewildered, at his own palms – Harry could see they looked burnt, raw, red and shiny.
‘Then kill him, fool, and be done!’ screeched Voldemort. Quirrell raised his hand to perform a deadly curse, but Harry, by instinct, reached up and grabbed Quirrell’s face –
‘AAAARGH!” - Harry Potter and the Philosopher's Stone, Chapter 17 (The Man with Two Faces)
It's also not clear whether Quirrell gained any of the Dark Lord's magical abilities.
The only spell Quirrell is actually shown doing is the nonverbal spell he uses to tie Harry up. That would be a nonverbal conjuration, which is fairly difficult magic, but not so difficult that only wizards as powerful as the Dark Lord could do it. In addition, it's also never really mentioned how skilled a wizard Quirrell is on his own. It's never said whether Quirrell does any spells in class, and it's certainly not said that he did any extraordinary feats of magical power.
Only in the movie, Quirrell is shown flying, a power the Dark Lord is known to have. This would imply - but not conclusively prove - that his ability to do magic is enhanced by the Dark Lord's presence.
There's not much to go on, canonically. His powers might have been boosted by Voldemort, certainly—but there is no evidence of that.
However, he did suffer a few concrete effects from Voldemort's presence.
Voldemort's presence physically weakened him.
Quirrell is, in effect, turned into a temporary Horcrux by Voldemort. He is greatly depleted by the physical strain of fighting the far stronger, evil soul inside him.
Quirrell was made more vulnerable to protective magic.
Quirrell, full of hatred, greed, and ambition, sharing his soul with Voldemort, could not touch you for this reason. It was agony to touch a person marked by something so good."
—Harry Potter and the Sorcerer's Stone
One can surmise that his powers were amplified by Lord Voldemort, or at the very least shared. For example, Quirrell cannot determine whether or not Harry is lying about possessing the Stone, but Voldemort can - which points to mental abilities not being shared.
Physical abilities, on the other hand (that is to say, things that affect motion of the body or the physical wandwork required to perform, say, Stupefy), seem to be shared.
For example, Quirrell can fly when possessed by Voldemort, something out of the norm for most wizards, and an ability Voldemort himself displayed in the Battle of the Seven Potters.
As a second example, Quirrell is able to wandlessly and nonverbally conjure ropes and bind them to Harry. Conjuration is regarded (by Professor McGonagall) as an incredibly difficult form of Transfiguration, and as a mediocre wizard, he would have needed Voldemort's help to bind Harry.