Hey there!
Content-wise, the sentences mean the same. There is a difference, though:
I don't want to wear them. - suggests that the person talking is the one who is supposed to wear the shoes
I wouldn't want to wear them. - means that the person talking is not the one who is supposed to wear the shoes
wouldn't is used in an if-statement. You could add "[If I were you,] I wouldn't want to wear them" and it would mean the same. Because, in your example, the boy isn't the one who's supposed to wear the shoes the girl bought, he uses "wouldn't" instead of "don't". If she had bought the shoes for him, though, he would reply "I don't want to wear them."
I hope this helps!
Best regards,
Dana