The conflicts in both plays are totally different, because both plays are written in different time and, because they belong to different Genres (Drama, and Comedy) and they have been written by authors from different countries.
The conflicts in "The Importance of Being Earnest", on the one hand, are very superficial and are not meant to be far reaching, because they should amuse the reader. .
Due to this the characters in "The Importance of Being Earnest" are not developed very deep, because there are no deep conflicts. .
Most of the comic in "The Importance of Being Earnest" even is produced by the conflicts. They are even absurd.
The conflicts in both plays had to be so different, because in the case of "The Importance of Being Earnest" Oscar Wilde tried to criticize the people from the society, which mainly was watching his play, and by bringing so much comedy into the conflicts he managed to make his audience not feel appealed by the play.
On the other hand, are the conflicts in "A Streetcar Named Desire" more far reaching and the characters are more developed, this is logically, because "A Streetcar Named Desire" is a Drama and wants to make the reader think about he reads.
One can recognize that by analyzing the conflicts in both plays in further detail.
In "The Importance of Being Earnest" the conflicts are often presented very jokingly, this can sometimes give the reader/spectator the impression that they are not very serious.
But in "A Streetcar Named Desire" the reader/spectator can notice how the characters are really caught by these conflicts, like, for instance, Stella, who is caught between the two lifestyles, by the aristocratic one she as grown up with, for which Blanche is standing for, and by the lifestyle of workers, for which Stanley is standing for and which she has chosen to life.
Therefore Stella is also caught between the two main characters Stanley and Stella.
Also the reader can not notice such a complexity of conflicts in the play "The Importance of Being Earnest".