A braggart is someone that talks a lot about his or her own accomplishments in a way that will make him or her look better to others. It can also be an adjective with the same spelling.
Bragger is the noun form of brag, which is to talk about one’s self in a boastful way that will make one look better to others.
In other words, these terms are synonyms with braggart being the more popular, used twice as much as bragger. Some would make the distinction that being a braggart is more offensive than being a bragger and should be reserved for those who are truly atrocious in their self-proclamations.
Examples
It’s a strategy smartly followed by filmmaker Kelly Reichardt for her tale of a small wagon train of Oregon settlers led by a braggart wilderness guide who may be getting them increasingly lost or leading them into danger. [Indie Wire]
Listen to Rosemary: “When I met John (not his name) I did not like him at all. My impression of him ‘was a real braggart’ and to make matters worse, I had heard so much about him before even setting my eyes on him, but nothing good. [The Guardian Nigeria]
Fly fishermen are equal parts discreet and braggarts (another column entirely…). [Reno Gazette Journal]
The bracket bragger is just like the guy who will spare no details when telling you about his fantasy football team in the fall. [Huffington Post]
Pointing to the decrease in unemployment since he took office in January 2011, Scott said Florida is “going to become better braggers than Texas about how we’re getting more jobs.” [CNN]