Why does Captain Jack Sparrow suffer from the curse?

Actually Captain Jack Sparrow didn't get cursed when his old crew found the Aztec gold.

What you may be thinking of is the point in the movie where Jack becomes undead for short time. In the cave, while Barbossa and Jack negotiated, Jack stole one Aztec coin from the chest getting himself cursed until he returned it.

Here you can read fairly detailed plot in case you missed something.

He wasn't cursed, to begin with, it was only near the end of the film when he takes a coin directly from the chest that he gets cursed.

Jack was not cursed, to begin with, but once he persuaded Barbossa to ally himself and not kill Will, he picked up 4 coins and threw back 3 to dramatically say "every" (throws down coin) "last" (throws down coin) "One" (throws down coin). This was a trick to make It seem like he replaced all coin directly back to the chest without really taking any BUT he had a fourth coin and the camera shows him slyly palming it and extending his fingers to indicate an empty hand, classic gamblers trick. He then stepped away from the chest which in turn would curse him. Another factor to this plan working is that the rest of the pirates loathe the curse having been under it for so long, It probably didn't occur to them that someone would deliberately allow themselves to be cursed just to be temporarily unkillable.

The main reason Jack did this isn't necessarily that he was worried he'd die in general but he knew he'd have to fight Barbossa who has been suggested to be even better than him at swordplay.

To answer some above suggestions that Jack was Immortal since before then, no he wasn't.

One of the crew saying "you're supposed to be dead" was In reference to him being left on an Island to die but only Jack knew It was a regularly stocked point for merchants. No doubt knowledge from his days working for the East India Trading company and Jack Is known to keep his cards close to his chest.

The apple he ate wasn't finished, Barbossa took it before he had Jack locked up. Barbossa inspected the apple and angrily threw it away. This, coupled with the fact Jack rummaged through the apples on the table suggests that Barbossa isn't in the habit of poisoning all of the apples, only the top ones, and Jack knows his methods having been crewmates and friends.

Additional note: Jack and Barbossa are solid friends but It is likely the curse made Barbossa unusually ruthless to get what he wanted. He said he felt nothing, what would this say of his conscience? There probably wouldn't be one, hence why when he was alive again he was far more amicable with Jack, had his honour and "sense of fair play".