Having a look on dictionary definitions:
Someone who is boastful often talks proudly about things they have done or can do, or about something they own, especially in order to make other people admire them
Someone who is immodest thinks they are very good or clever and likes to talk about their achievements.
I hate the way she behaves. She often always speaks about her husband and his wealth and his influence in the gold market and the number of properties they own. She has never had any new word to say. She's really a/an __________ person.
I see nothing different between these two adjectives in this sense and I think I can interchange them here in this specific example.
I need to know how a native speaker thinks about them.