Unlike previous generations of TIE fighters, the Special Forces TIE/sf actually comes equipped with a compact shield generator.
image from Star Wars: The Force Awakens - Incredible Cross-sections
It would appear that Poe Dameron was able to activate this shield before the ship was hit with the blaster fire from the bay. It's mentioned slightly later on in the film's Official Novelisation (when they take out the Star Destroyer's lasers) that the shields are active.
The whole gun emplacement erupted in a rapidly shrinking fireball.
Debris spun around them as Poe took them through the devastation, the
fighter’s shields warding off whatever he could not directly avoid.
By comparison, there's no indication that Poe had activated the X-Wing's shields. He was trying to avoid being detected by scanners until he was in a position to attempt a quick takeoff. That presumably would include not activating systems with large electromagnetic signatures that could be tracked, such as the shields and main weapons.
The ship that was parked some distance from the village was well
hidden beneath a high rock outcropping. That wouldn’t shield the
X-wing from sophisticated search gear, Poe knew. He needed to exit the
atmosphere, and fast.