It's not Snape that triggers the pain.
Professor Quirrell, in his absurd turban, was talking to a teacher
with greasy black hair, a hooked nose, and sallow skin. It happened
very suddenly. The hook-nosed teacher looked past Quirrell's turban
straight into Harry's eyes -- and a sharp, hot pain shot across the
scar on Harry's forehead
And guess who's in the turban...
Some clarification regarding Harry's scar not hurting at other times he meets Quirrell.
First, Voldemort needs to be feeling strong emotion - this is confirmed in later books by Dumbledore. Secondly, while he is weakened and in spirit form, proximity is also necessary, hence his scar only hurting in the direct presence of Voldemort pre-GOF.
The events in this book that cause Harry's scar to hurt all involve scenarios in which we can imagine Voldemort is feeling a violent emotional jolt.
The scene the questioner is asking about is the first time Voldemort has "seen" Harry in person since he tried to kill him. Naturally, you can imagine he would feel some anger/emotional upheaval. When Harry stumbles across him in the forest Voldemort/Quirrell moves towards him, as if to attack - strong bloodlust/anger - a chance to kill the boy who lived! Finally, on physical contact with Quirrell in the finale - extreme pain caused by the protection curse. Probably some fear as well.