Running byobu will launch with your selected default backend. You can choose between tmux and screen by running byobu-select-backend.
Select the byobu backend:
Choose 1-2 : 1
I think what you want is to launch the Byobu desktop application. You can find that in the Ubuntu application launcher, by simply typing Byobu. You can lock it to the launcher, and the icon will always stay there, on the left.
This is a little writeup on how to get a very useful terminal setup on any Debian based Linux. Example of my setup below: