Jem cries in chapter 22 because he feels a great injustice has been done to Tom. Jem believes with all of his heart that Tom will be acquitted, so when the guilty verdict comes down, Jem is upset. He's so upset, angered, and saddened by what he sees as an injustice that he breaks down and cries. Atticus cries for a similar reason. Atticus holds it together, but when the black community shows their love and support to the Finch family by leaving a ton of food for them Atticus begins to cry. He cries because he too is saddened and angered at having lost the case, but it is the love he receives, despite losing, that causes him to finally break down.
Dill and Scout cry earlier in the novel, for many of the same reasons that Jem and Atticus cry. Both Dill and Scout have tender hearts (not weak). They are capable of experiencing and expressing empathy for a great many people. In the case of this novel, that empathy is greatly applied to the African American members of the town. That makes the Finch family and Dill quite unique in the novel (and this time period in history).
Mayella cries because she is an abused character. She is likely beaten by her family. She has the mental capacity of a child much younger than her. She is prepared to lie under oath, but Atticus systematically breaks down her responses until she is so distraught that she breaks down, cries, and then refuses to answer any more questions. She cries, yes, but the reader feels very little sympathy for her, because we know that she inflicts (or tries to inflict) more harm on others than she receives. That lack of sympathy from the reader and her general reason for crying separate Scout, Jem, Dill, and Atticus from Mayella. The Finch family has a heart for justice and cries when it is not given. Mayella cries because she was caught in a lie.