In the short story "The Necklace" by Guy De Maupassant, the author also begins with a dilemma. This dilemma is whether Mathilde Loisel should go to the Ball or not. Clearly she hovers and vacillates. If she doesn't go, she misses out on probably the only social invitation she might be fortunate enough to get all year. If she does go, she will have to take risks (spend money on a ball gown/acquire by hook or by crook some fancy jewelry.) She appears to use her dilemma to put pressure on her husband saying she cannot (will not?) go without new finery. He feels guilty as he has a little money saved. They do not realize how lucky they are (a job,a roof,food and even a little money saved.) the consequences where they lose all that are the result of taking the wrong road out of Mathilde's dilemma.That leads to more dilemmas and a serious predicament.
Concerning your question about "The Necklace," there may be a philosophical or other definition of these two words that I'm not aware of. Using dictionary definitions, though, Mathilde's situation is probably more of a dilemma than a predicament.
A predicament is a difficult or perplexing or trying situation. A dilemma is a usually undesirable or unpleasant choice. There's a fine line between the two, and I believe I could make an argument either way. But for the sake of an assignment, I'd choose dilemma.
Mathilde faces an unpleasant choice. Neither option available to her is a good one. She can tell her friend that she lost the necklace and face, she thinks, humiliation, or she can borrow everything possible, cut expenses, and work endlessly for years to pay the money back. Her future is dependent on her decision.
I believe predicament would be more of a situation than a decision. Something like getting a flat tire in the middle of a desert, or something like that. I think you're safe going with dilemma.
I find that Mathilde's situation in Guy de Maupassant's short story, "The Necklace," is both a dilemma and a predicament.
Dilemma: A situation that requires a choice between options that are or seem equally unfavorable or mutually exclusive.
Predicament: A situation, especially an unpleasant, troublesome, or trying one, from which extrication is difficult.
Certainly, Mathilde has a dilemma once she discovers the necklace missing: Does she tell her friend the truth or return a similar item (even though it may cost the family everything it owns)? The lost necklace has also left her in a predicament which is highly difficult to solve. It would be difficult to choose which term is closer to Mathilde's situation, but both seems to fit her problem accurately.