In "Miss Brill" an external character vs. character conflict causes an internal conflict.

Conflict is a struggle.  There are two main types of conflict: internal and external.  An internal conflict is a struggle a character has with his or her self, and an external conflict is a struggle a character has with another.

As in many cases, an external conflict causes an internal conflict in this story.  Miss Brill is enjoying her time in the park, and sees herself as an integral piece of the play.  She is proud of her fur, and appreciates the interactions.

And what they played was warm, sunny, yet there was just a faint chill no, not sadness—a something that made you want to sing.

The external conflict comes from when Miss Brill overhears two young people insulting her.

"But why? Because of that stupid old thing at the end there?" asked the boy. "Why does she come here at all—who wants her? Why doesn't she keep her silly old mug at home?"

She realizes then that these characters do not want her.  This is a character versus character conflict.  The other characters are mean to her, even if they don’t know that she can hear them.

This produces an internal conflict.   Miss Brill feels bad.  She begins to see herself as a burden, and returns to the isolation of her dark and lonely home.

But to-day she passed the baker's by, climbed the stairs, went into the little dark room her room like a cupboard——and sat down on the red eiderdown. She sat there for a long time.

Miss Brill feels lonely and regretful.  She no longer sees herself as part of the great play taking place in the park.  She now sees herself as isolated from the play she used to be a part of.  She imagines she hears crying.  This is a result of the internal conflict she feels.

The story demonstrates how one person’s hurtful words can affect another person so completely as to change that person’s self-perception.  The young couple have no real interactions with Miss Brill.  There is no reason for them not to like her.  Yet they think she is ugly and silly.