There is an option in your profile settings to show name or username.
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Extending on jormond's information, if you have your Name set to "Yourself" in your Profile's Privacy Settings only you will see your full name. This is all modifiable from the Menu in the Top Right > Edit Profile > Privacy...
I hope this helps!
When you are logged in, your screen name is your full name (to you). And when not it's your username. My name does the same.
If you want everyone to see your name, under your avatar, select Edit Profile and then the Privacy tab. Next to "Name:", change the selection menu.