terminal in Tmux is 256color but vim isn't

Recently I installed tmux on my Ubuntu 16.04. The terminal screen in tmux shows 256 color correctly however when I opened vim within tmux, the color was different from what it showed when it was run directly on the terminal. I tried to add set -g default-terminal "screen-256color" in my .tmux.conf file but the problem still exists. * my terminal screen in tmux:  * vim run directly on the terminal:  * vim run within tmux:  The output of echo $TERM; tput colors; tput longname * outside tmux:

xterm-256color
256
xterm

* with 256 colors inside tmux:

screen-256color
256
GUN Screen with 256 colors