Simo Häyhä was an anonymous figure in a 105-day war where the Soviets bungled up fairly massively, while Törni (AKA Larry Thorne) was the relatively well-known commander of hand-picked reconnaissance and strike company, "Detachment Törni," operating from 1943 to 1944. Törni and his men weren't shy of occasionally advertising their achievements (which were genuine, to be fair), and plenty of troops in the area where Törni operated were aware of the reputation they had. (I've read of couple occasions when the arrival of Törni's men - who were also fairly often used as reserve force - were greeted by sayings such as "Törni's jaegers are here, NOW things are going to happen.") From prisoners and from other intelligence means the Soviets were able to identify the man who vexed them so.

Simo Häyhä's exploits have become widely known only here in the Internet. I, for one, learned of them from the Internet even though I've been trained in long range reconnaissance in the Finnish Army. (Törni's operations on the other hand were discussed, and there was even a Camp Törni in Kosovo at one point - still have the "lightning T" shoulder badge that was copied from Detachment Törni's original insignia.) Häyhä's exploits have also been highly embellished and it's fairly likely a lot of things didn't go as some of the more popular English accounts claim they did. /u/Elm11, /u/Mosinista and I have a plan to provide a better overview of the man and his achievements (which were considerable in reality too) some time in the future.

Sources:

e.g. Cleverley's "Born a Soldier: The Times and Life of Larry A Thorne."