A monolingual dictionary can often help with this kind of question. Merriam-Webster, American Heritage, and Chambers are all easily available online, or here are the definitions from the New Oxford American Dictionary (NOAD):
flinch - v. [intrans.] make a quick, nervous movement of the face or body as an instinctive reaction to fear or pain: 'she flinched at the acidity in his voice' | 'he had faced death without flinching.'
(flinch from) [figurative] avoid doing or becoming involved in (something) through fear or anxiety: 'I rarely flinch from a fight when I'm sure of myself.'
wince - v. [intrans.] give a slight involuntary grimace or shrinking movement of the body out of pain or distress: 'he winced at the disgust in her voice.'
I agree with Archfarchnad that literally, wincing is a pained facial expression involving mostly the eyes, whereas flinching is more a reaction of the whole body, especially a sudden or involuntary jerking away from something:
· The sudden bright light made her wince.
· The sudden sharp cold made him flinch.
· I could see him wince at the thought of having to rewrite the entire paper.
· Even though the snake was a tame, harmless pet, I could feel her flinch as her fingers made contact with its skin.
However, the line between them may not be absolute:
· He winced/flinched as the alcohol came into contact with the wound.
Of the two NOAD examples with '_____ed at the ____ in someone's voice,' the only difference seems to me that 'wince' is milder while 'flinch' is stronger, but unfortunately that's not very clear from context and the two sentences are probably largely interchangeable. Maybe the lexicographers just weren't at their best that day.
Figuratively, 'wince' often implies distaste or embarrassment, that is, pain in an aesthetic sense:
· His electric-blue pantsuit was garish enough, but the rhinestones really made us wince.
· She sang so loudly and so out of tune that the judges actually winced.
'Flinch' is often used in the negative to suggest unusual courage or fortitude:
· The blow almost knocked him off his feet, but the boy never flinched.
· The rope bridge swayed ominously above the rocky gorge, but she set foot on it without flinching.
· The bank robber didn't flinch at the thought of a 40-year prison sentence.
· The police detective didn't flinch from her duty, even though it meant arresting her husband on suspicion of murder.